What does John Calvin mean to imply by the "suffrage" of the church? In his Institutes of the Christian Religion, John Calvin discusses the idea that the Biblical canon depends on the ratifying action of the Church (Book 1, Chapter 7). He thinks it does not, preferring to say that the Church formally recognized what everybody already knew. This is meant not only as a historical statement about prevailing opinions, but also speaks to the role of the Holy Spirit in testifying to the authenticity and authority of Scripture.
The Latin and French texts use the term suffragio or suffrage respectively to describe the approval of the Church:

Quare dum illam recipit, ac suffragio suo obsignat, non ex dubia aut alioqui controversa authenticam reddit : sed quia veritatem esse agnoscit Dei sui, pro pietatis officio, nihil cunctando veneratur. (1559 ed.)
Parqouy l'Eglise en recevant l'Escriture saincte et la signant par son suffrage, ne la rend pas authentique, comme si auparavant elle eust esté douteuse ou en different : mais pource qu'elle la cognoist estre la pure verité de son Dieu, elle la revere et honnore comme elle y est tenue par le devoir de pieté. (1560 ed.)

English translators have rendered this language in the same sorts of ways as one another:

When the Church receives it, and gives it the stamp of her authority, she does not make that authentic which was otherwise doubtful or controverted, but, acknowledging it as the truth of God, she, as in duty bound, shows her reverence by an unhesitating assent. (Henry Beveridge)
Thus, while the church receives and gives its seal of approval to the Scriptures, it does not thereby render authentic what is otherwise doubtful or controversial.  But because the church recognizes Scripture to be the truth of its own God, as a pious duty it unhesitatingly venerates Scripture. (Ford L. Battles)
For that reason, when the church receives and approves scripture it does not authenticate it, as if it has previously been doubtful or unsure. But since, according to its duty, the church recognizes scripture to be the truth of its Lord, it reveres scripture without delay. (Elsie Anne McKee)

The Latin suffragio can indeed mean a vote, seal, etc. - some notion of a formally recorded act of testimony. The accompanying obsignat matches that.
What I am wondering about is the intended connotation of the &quot;seal&quot;-type metaphor and the word suffragio / suffrage. Elsewhere, seal language is used for sacraments, and the concept of our assent to God in faith seems very similar to what is implied here (we put our trust in God, responding to him, but we don't make him do anything).
With reference to Calvin's concept of the church,

Is suffrage here intended to denote some communal expression of faith? Or is it just that many individuals independently put their trust in the authority of Scripture? Basically, I'm asking to what extent &quot;the Church&quot; is present here as a single entity.
Is there a meaningful concept of voting? Clearly, English &quot;suffrage&quot; today probably makes us think about aggregating everyone's opinions and picking the most popular. But does Calvin intend to suggest anything like that? I think he is pointing to a vaguer notion of informal universal consensus, especially given that there is only one correct choice, but I'm not fully sure, given that some people are convinced Calvin was some sort of pioneer of democracy.
Is the parallel with Calvin's sacrament language just a coincidence, or are the same concepts being invoked in both instances?
Answer for question 1: Suffrage 

Suffragio also means intercession.So Suffragio can mean suffrage , vote, intercession or a combination.

With this understanding of suffrage, vote and intercession. 
Lets take your posting and compare within a context that Calvin may be referring to.

"Thus, while the church receives and gives its seal of approval to the Scriptures,
 it does not thereby render authentic what is otherwise doubtful or controversial. 
 But because the church recognizes Scripture to be the truth of its own God, 
 as a pious duty it unhesitatingly venerates Scripture."

According to the above statement, let's look at a perfect example. One of the first ancient councils where "Suffragio", intercession and vote was needed.

Example: 

The First Council of Nicea, was a case where The Church was under attack from
 heresies, Arians denying the divinity of Christ. 
 Similar to the statement above declares.
 "it does not thereby render authentic what is otherwise doubtful or controversial"

Likewise the Council of Nicea did not render authentication of what was doubtful or controversial. It was the opposite, that is to say
 The Church was defending Apostolic teaching, defending what had been passed down and always understood and believed. Defending what they already knew.
 No new theology came from this council.  Suffragio (intercession and vote) was needed because of outside heretical forces to defend the scripture.
 Thus the Nicene creed came into existence. Not creating anything new, but again defending what was already known, understood, believed and passed down 
 form the Apostles. Everything being defending through suffrage,intercession and vote.

This example would also answer your question about a "communal expression of faith". The ancient Church believed that as long as the entire Church was united,
 and voted united, that the Holy Spirit was guiding them. This is very evident throughout the first seven ecumenical councils where "Suffragio" was required. 
 In contrast, if any one part were to leave, there could no longer be ecumenical intercession or votes. A foot note that ever since the Great Schism, Rome being separated from 
 the main body of the other four (Alexandria,Jerusalem,Antioch,Constantinople), there has never been another ecumenical council in the East, that included all.
 Rome continued with their own councils only effecting the West, for which eventually brought about Luther's and Calvin's writings..

You next inquiry is:
 "I'm asking to what extent "the Church" is present here as a single entity."
 This is a great question and a little complex depending on context. Most of Calvin's writings are against the Roman Church in some way or another.
 He may very well be making his statement in context of all other historical occurrences and examples before the Great Schism, in rebuttal to the Roman Church's changes
 in the West after the Schism. If he is writing in the context of Sola Scriptora, this would be a difficult position to defend, as other Protestant churches
 had already come into existence and there is no logical way he could pick from one of the many diverse entities who did not agree, claiming a single entity.
 The concept of an all encompassing invisible Church does not exist anywhere in Christian history before this. The Church is very visible and united for over a 1,000 years until the Great Schism.

Let's explore all possibilities he could be thinking.
 So "the Church" is present here as a single entity" would depend on context.


The original ancient Church that stood united for over 1,000 years is considered The Church.
After the Great Schism of 1054 AD, Rome claimed to be The Church.
Likewise The Church of the East, the Orthodox (Alexandria,Jerusalem,Antioch,Constantinople and Rome) claimed to be The Church and still claims this.
According to translation Calvin is using a present tense. "recognizes" which alludes to a more present understanding or what Calvin's understanding of "The Church".

If the case is option 4, this would allude to an understanding only by Calvin. Basically his own understanding.

Note: If the translation was in a past tense. "recognized",
ex:'But because the church recognized Scripture to be the truth of its own God'
I would say he would have been referring to past occurrences, when The Church was united. Maybe again as a rebuttal, to say Romes intercession's , votes were not in defense,
but rather they were changing things. He still could be meaning this in one form or another.  


Answer for question 2:Is there a meaningful concept of voting?
   In accordance to Church history, Yes! However, only when The Church (ALL) is fully united. Thus the reason there were never more than
   the seven ecumenical councils. Rome being separated in 1054AD, to this day, no other ecumenical council has ever taken place. Not because Rome was more important,
   Each bishop whether from Rome or Jerusalem was allowed only a vote a piece, it's more along the lines of The Church has been separated. 
   I would suggest looking at the Ante-Nicene collection and the councils specifically to help you understand the need for the councils and voting in ancient times.

I would not think at all that Calvin purposely intends "aggregating everyone's opinions and picking the most popular". That would not match anything in Christian history.
   However he is in a little bit of a dilemma, writing against the Roman Church, but not being united with the Church from the East.
   All Churches will say the are being lead by the Holy Spirit, but don't agree.I personally think Calvin has a deeper understanding, however the position that he is in, the era, his surroundings, no contact with the East, is indeed a difficult one. 

Answer for question 3: Is the parallel with Calvin's sacrament language just a coincidence, or are the same concepts being invoked in both instances?
  The sacramental language is not a coincidence. Again, based on history, the seven ecumenical councils of the East are considered Holy occurrences of Suffragio and defense within The Church.  I'm very confident in my answer to say that Calvin, within his own understanding, is expressing a sacramental expression. I will simply express again, within his own understanding.