I'm trying to write a server for a chat program. I want the server to have a tcp connection with every chat user. Is there way for the server to have multiple tcp connections at the same time without creating a socket for every connection? And if yes, how?
No. Unlike UDP sockets, TCP sockets work are connection-oriented. Whatever data is written into a socket, "magically" appears to come out of the socket at the other end as a stream of data. For that, both sockets maintain a virtual connection, a state. Among other things, the state defines both endpoints of the conntection - IP and port numbers of both sockets. So a single TCP socket can only talk to a single TCP socket at the other end.
UDP sockets on the other hand operate on a per-packet basis (connectionless), allowing you to send a receive packets to/from any destination using the same socket. However, UDP does not guarantee reliability and in-order delivery. 
Btw, your question has nothing to do with python. All sockets (except raw sockets) are operating system sockets that work in the same way in all languages.
Multiple TCP connections in python