In English there is not a focus on explaining the amount of experience a peer has like words in asian languages do. Most native English speakers would use classmate for the all university cases. Contemporary and peer are synonyms. Contemporary is more general and formal and peer is more specific to what you are describing. There is also co-worker or colleague for working situations. Hope that answers your question and good luck with your studies.

For University, use classmate. For Company, use co-worker. Neither applies to "entering in the same year" but can be used in more diverse situations. Example: "My co-worker had been here 3 years and is still teaching me the ropes of the job." "Even though Anna was a senior, she still took a junior level class so she is my classmate." Or if you mean literal calling them, just use their name... Example: "Hi. Can we discuss yesterday's reports, Mr.Smith?" "Hey. How are you today, Angie?" "Hey, classmate, what was your name again?" "Yo, we're classmates right? "That's Sam, my co-worker."

Contemporary is the word we use, yes. It is a formal word, not likely to be heard in everyday speech. They were contemporaries; they both worked for Google at the same time but the never met.

I, personally, wouldn't use "contemporary" over "peer", but generally, a contemporary is someone who is kind of like a peer, but it implies that you are equal in some way (work ethics, knowledge, et cetera). A peer is more someone who has equal standing in, say, a classroom or working environment. I hope I helped!