No, usernames are not reclaimed. Those names may be tied to site content or games. Sometimes people go away and eventually come back.
Freeing up old usernames

However, you might want to submit a ticket: https://support.chess.com/customer/portal/emails/new and inquire about the possibility of claiming the name you want. You never know for sure until you ask. Stranger things have happened.
When registering an account on chess.com, my usual username was already taken. Today I decided to look into it and saw that the person who has that name hasn't logged on in over 6 years. Does chess.com ever clear out unused accounts to free up usernames? I've started to use the site quite a bit, and I'd love it if I could use my typical user name here.
Thanks!