On chess.com correspondence games are organised in forums. Could it be made possible so that people can play each other in threads which are labeled with a certain tag?
Technically, yes. But that's not what Chess Stack Exchange aims to be:
Chess Stack Exchange is a question and answer site for serious players and enthusiasts of chess. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about chess.
Indeed, a separate tag for those 'questions' would be a good way to separate them from regular questions, and make it possible for users who are not interested in them to hide them. Note that by default they are visible, and non-registered users don't have that option, so if there are a lot of active games, they won't see regular questions. Most of our traffic is actually non-registered users, so that's my main concern against this plan.
Also, you would need to sort answers by posting date, not by votes (which is the default). I'd imagine good moves get upvoted more often, so reputation would be a mix of Stack Exchange reputation and ELO rating. (Though these posts could be Community Wiki, but that feels weird too.) That means for instance that good chess players who know absolutely nothing about Stack Exchange would get to vote to close questions.
All in all, it's an interesting thought experiment but it's not the niche Stack Exchange wants to be in. One of the advantages of the Internet is that there can be several websites about the same topic, but each having their own role or specializations.