In Telegram if I can convince the phone network that I have your phone number, Telegram will deliver all your Telegram messages (including historical ones) to me and I can read them. This is completely transparent to the other participants.

In Signal the remote Signal server will not deliver the messages, since anyway I can't decrypt them (there are no unencrypted messages in Signal) and it never keeps any that it has successfully delivered.

Now, if I convince the network that I have your phone number, how should your friends determine that I'm not you?

In Signal (and WhatsApp) this is designed into the UI. If you aren't sure if this is who you think it is, you can meet in person and compare numbers or QR codes to check there are no shenanigans going on. Many casual users never do this, but if something seems "off" you can verify.

But in Telegram there's no option like that.