According to Christianity, is only the Christian faith a gift from God? At least in the Catholic Church (with which I am most familiar), it is stated that Faith is a gift from God. Point 179 in the Catechism reads:


  
  Faith is a supernatural gift from God. In order to believe, man needs the interior helps of the Holy Spirit.
  


A related question has an accepted answer on the same lines.

Now, we know that people from other (even extinct) religions also have faith in their particular god(s). Within those religions, some have more faith than others, and there is also a group which have lost their faith, perhaps because of conversion to a different faith.

According to Christianity, is their faith also a gift from God? Why would God give to some a partial/incorrect gift and a complete/correct gift to others? What is the Catholic (and other Christian) teaching on this respect?

P.S. This question emerged in the context of a conversation with a Muslim friend. Some of them also believe that faith is a blessing from God (e.g. here). Hence the question of the many but different gifts.
Most of Christianity –including Catholicism and Protestantism¹– makes a distinction between saving faith vs. generic faith. One can have faith in your toaster's ability to evenly heat your toast or faith in the laws of physics but these kinds of faith don't make you right with God. As you pointed out even other religions have things they call "faith", and even godless religions such as humanism or atheism have their own kinds of faith.

What Christianity says is that saving faith in Christ that is our only path to being reconciled with God cannot come through our own impetus but must come through God's Spirit acting in us.

As such all other "faith" is empty — no other "faith" saves us. Neither faith in any specific religious system or generic faith not placed in anything specific will be instrumental in saving men from their sins. Hence nearly all Christians would say that these other "faiths" are not gifts from God.

How exactly they are described varies some between traditions: some blame the Devil, others say it's just man's sinful nature conjuring up counterfeits. But the result is largely the same — saving faith in Christ is a gift from God, not any other kind of faith.

Where different Christian traditions differ on this issue is on how much of the responsibility for this faith lies on man and how much on God. As an example contrasting Catholicism and Reformed Protestantism, the former would say that God gives the gift of faith rather broadly making it available to men but men have to pick up the ball and run with it while the latter would say that saving faith can only ever be exercised by a heart that has been regenerated by God and that living out that faith is the inevitable consequence of having been granted such a faith.

It probably also bears mentioning that faith in the Christian sense is not some nebulous abstract. It is concrete in that we know exactly what we hope for (eternal communion in the presence of God), by what means we expect to realize that hope (Christ's sacrifice for sins reconciling us to God and adopting us as his own children), and have been given concrete evidence that what is promised will be delivered (the history of redemption through the OT, the resurrection of Christ in the NT, the Holy Spirit with us now).

Virtually all other kinds of so called "faith" are much more abstract — either nebulous or blind. You mention Islam for one. I live in the Middle East and am quite familiar with Islam's version of faith and it is not so different from a humanists hope in humanity. To a Muslim faith is a tenuous hope against hope that if they do enough of the right things Allah might  "mercifully" overlook their faults when the time comes. There is no assurance that he will, much less a guarantee that he has (as in Christianity) only a vague hope that he might forgive them. To a deist or agnostic who has "faith", its just so nebulous that it hardly means anything: faith to my friends in this category is a tenuous prayer into the void hoping everything will all work out in the end, but they haven't a clue how or why it would. Christianity, on the other hand, is very definite about how things will work out, who they will work out for, and why it will be that way. Faith for a Christian is foreknowledge of a certain outcome based on reliable proof in the past. 

¹ Orthodoxy is a little harder to pin down on this issue because it downplays the role of faith in salvation at all. Some liberal mainline Protestant denominations would also soft peddle this issue as they are not firm on Salvation being found in Christ alone.