What's with the serial number for each game? Are they required for playing just the single player (I know that there's multiplayer but nobody cares about it)?

Can the serial be used unlimited times on different computers or do they have limited use?

The serial number is not needed to play single player.

Refer to this thread here:
http://www.gog.com/forum/stalker_series/settings_program_has_a_cd_key_field_in_stalker_shadow_of_chernobyl

No problem. Now, go and blast away some bandits and mutants!

I know your question has already been answered but I just wanted to add: don't worry whenever you see something about a serial in a GOG release. There's quite a few games on GOG that used to require a serial in the original retail version and most of the time you will find some leftovers of that, it does not ever lead to an installation limit or anything, GOG wouldn't allow for that kind of thing.