It is neither against the signal software’s license, nor it is against the signal service’s terms of service.

This is a false meme spread because the Signal founder (who is no longer with the company) didn’t like people making forks without changing the API server URL and running their own servers.

Open source software doesn’t work like that, however.

Whether they’re open source doesn’t matter (for this question). They control (their instance of) the server.

As you say, I do remember them issuing some threats about it, so it would be interesting to know if they’ve changed their stance on this.

(Discord, as an example, has banned users for using alternative clients.)

Alternative clients are banned in the Discord TOS. The Signal TOS is on their website and doesn’t prohibit any clients.

Also, separately, the idea that you can only use a service with a certain client is dumb.

Imagine if a website said you can only use a certain browser, or they ban you. It’s ridiculous.