In short, there's no provision for a signature in the SMS protocol, so the app has no place to add one. (There's no signature header in emails either, but since everyone uses a cut line [well ... everyone who knows how to do it properly - a line with just -- then the signature on the next line, and no more than 3 lines of signature], most email programs have been written to be able to handle this.)
When RCS becomes the default method of handling texts, there's probably going to be a signature header or method included in the protocol. (There's nothing in the protocol now, but RCS is still in diapers.)