Yes, but in my limited experience it's been former managers who are looking to see if they can re-hire you (at either your previous job or wherever they moved onto themselves).

I think that’s just selection bias. Managers became mangers because they’re the type of person that keeps in contact with their prior coworkers and associates.

I'm not really a birthday person in general, so no, not that specific thing, but I definitely have a set of like five to ten past coworkers who I really respected and enjoyed working with, some as peers and some as managers and mentors, and maybe about once a year I'll think "I wonder how XYZ is doing" and reach to them.