There is no exact English translation of the Russian word "ровесник." As a previous poster said, we usually say "this person is the same age as me" or "we're the same age," etc.

I think the word you are looking for is "peers".

You could say "contemporaries", which would be standard language.

"People my age" works, sort of. In a formal setting you might talk about a cohort, or an age group. But I've never heard "age mate" :( I've heard a rumour that US English people talk about the "class of 2010", or whatever year, relating to when the group left high school. But I don't know if that's true. ADDITION: torusan's suggestion is excellent, as its Anton's. Oh, and someone above here suggests "my generation" - which was the title of an old Who song :) I'm obviously asleep today ... https://youtu.be/qN5zw04WxCc

No, we don't say "age mate." I actually don't think we have a word for this - we'd usually just say something like "this person and I are the same age."