TLS always authenticates the server (except in some variants that are never used on the web because they're only useful in very specific environments). This is necessary because if neither party authenticates the other, then they cannot ensure that they have an encrypted connection, as opposed to each having an encrypted connection to a man-in-the-middle who decodes and forwards the traffic. The server in an HTTP connection is not anonymous anyway.
TLS only authenticates the client if the client offers to be authenticated. This feature is not commonly used on the public web, mostly on intranets, and more commonly with protocols other than HTTP (for example email clients). Your client is not going to accidentally send a certificate that authenticates it: it would have to create a certificate first, and it would typically not use the same certificate with another client. Of course if you use a client program that you don't trust, it may reveal information about you, but doing it by creating and using a client certificate is a very unlikely method.
TLS itself doesn't send any directly identifying information about the client, but like with any other layer of the protocol it can reveal information indirectly through fingerprinting. The fingerprinting of a TLS client is mostly about which features it advertises to the server (which cryptographic algorithms and optional protocol features it supports).
For anonymity, your best bet is to stick with the Tor browser in its default configuration, so that your fingerprint matches the fingerprint of everyone else using the same configuration. If you use another client, then sticking to that client's default will likewise usually minimize the information leaked through fingerprinting.