As time goes by more and more games are being released for Linux and even more people start to play games on Linux. When I check for games on Steam the ones that can be played on Linux say just that i.e. to be played on Linux and not a specific distro. As there are so many of them and as even more are being developed constantly it seems that if a game can't be played on every distro then there is going to be a problem as every person chooses a different one based on his needs. Also as the "market share" is divided among so many distros then it will be very difficult to develop games for Linux. So my question is, does a game for Linux plays on all of the available distros?
No, Linux games certainly do not run on every distro. I'm confident, for example, that none of them will run under Debian Potato on DEC Alpha. While that's a silly example, it illustrates that Linux runs a lot of places, and has for a long time. More reasonable questions would be, e.g., ARM (e.g., for a Raspberry Pi). Again, the answer is probably not.
There is a useful answer, though: Steam documents their supported Linux distros, presumably that's what a game that supports Linux is tested to run on. I suspect Steam needs to update that list, though (odd they're still on 12.04 LTS).
The further away from the supported distro yours is, the more likely you are to run in to issues, of course.