Welcome to Android Central! This is typically not possible on SMS messaging apps. Think about it -- having a signature on every single text you send would quickly get very tedious and cluttered for the recipient. Signatures are more appropriate for emails.
Some text messaging apps allow setting a signature (unfortunately) - like Verizon's Message+ app. However, please consider B. Diddy's comments. People generally do not want to see a signature on every txt they get from you.