Why do the names of some Chinese companies consist of "repeated" words? I'm referring to the repetition of names such as "Dangdang" and "Renren." Why is this the case?
If you compare Chinese with English, you will find a lot differences and similarities. Similarities may help you learn Chinese a little bit easily, while differences are the things you need to keep in mind and to get used to. 

Here are the ways how English/Chinese sentences are constructed.

English:  Letters ----------------- Words ---- Sentences

Chinese:  -------- Characters ---Words ---- Sentences

You may have noticed that I located Chinese characters somewhere between English letters and words. Each Chinese character has its own meanings, and we combine characters to make words (a lot words contain only one character though, like 我,你,他, etc.). Most of Chinese words (more than 80% maybe) have meanings related to the meaning of each character used in that word. For the rest words, their meanings have NOTHING to do with the meaning of each character in the word. Most of the words in this category are those translated from foreign languages by pronunciation, like 马达（motor），基因（gene）. There are a lot such words in English as well, like lychee （荔枝），ping-pong（乒乓）.

One important feature of Chinese is that most Chinese words are two-syllable words. I guess two-syllable words sound more rhythmic than single-syllable words.

In your example, "Dangdang" 当当 (probably from the website name 当当网) is just a two-syllable name. 当当 is also used as the mimetic word for bells.

"Renren" is 人人。人 means man/people/person/mankind etc. 人人 means everybody. Other same pattern words are: 天天（everyday）， 月月（every month），年年（every year），处处（everywhere），个个（each and every one）， etc. 

Repeating characters in the SAME WORD emphasize the meaning of that character, and also sound more rhythmic. Same idea behind all the 4-character words in the patter of AABB, like 时时刻刻，日日夜夜，分分秒秒，吹吹打打，来来去去，年年岁岁，熙熙攘攘，热热闹闹，高高兴兴，干干净净，整整齐齐，恭恭敬敬，花花草草，战战兢兢，形形色色，从从容容，利利索索 and a lot others.

However, the above explanation doesn't apply to the repeating characters NOT IN THE SAME WORD. For example, 银行行长。银行 is a word, meaning bank. 行长 is another word, meaning the head of the bank.

Another example: 小明明天去上学。小明 is a name （probably a boy）. 明天 is a word, meaning tomorrow. While in 明明天天去上学, 明明 is a name （probably a boy）. 天天 is a word, meaning everyday.

It is very common in Chinese that repeating characters are used in the same word. Here are some cases where repeating characters appear.


in person's names (usually children's): 明明,亮亮,云云,圆圆,皮皮, etc.
in the words for simulating sounds made by things: 当当,轰轰,呜呜,呼呼,啪啪, etc.
in the words for simulating sounds made by animals:汪汪,喵喵,咩咩,哞哞, etc.
in the form of address: 姥姥,爷爷,奶奶,伯伯,叔叔,爸爸,妈妈,哥哥,姐姐,弟弟,妹妹,姑姑,舅舅, etc.
in the words in v+obj pattern: 尿尿 (小刚去厕所尿尿)
in the words in v+adv pattern: 开开 (请把门开开)
in actions: 看看,听听,瞧瞧,打打,写写,说说,读读,练练,弹弹, etc.
in words meaning "every X": 天天,处处,个个,年年, etc.
in ancient Chinese: 雨雨,将将,君君臣臣父父子子, etc. In this case, AA is a word, and usually the 1st A is a verb, and the 2nd A is a noun.


Sometimes, you have to use the words consisted of repeating characters, and it can't be replaced by the word containing only one character, for exmaple:

这是我的奶奶. Here 奶奶 is grandmother (Dad's mom).

这是我的奶. This sentence is totally different from the previous one. Depending on the context, this sentence can mean " this is my milk" when the speaker is having his/her breakfast, or it can mean "this is the milk from my breasts" when a mom breastfeeding her baby is talking to another mom.

If you regard Chinese characters as something similar to English words, you will be surprised to see repeating characters. However if you think characters as something similar to English syllables, then you will feel a lot comfortable.

Here are some English words that have repeating syllables:
mama,papa,yoyo,banana,Kijiji,Honolulu,Mississippi,Lady Gaga,Paganini,poopoo, etc.