Why would I have to give child support if I'm already supporting both, the child and the mother?

At this point keep receipts for EVERYTHING you spend. She will, one day real soon, take you to court for child support. Unless you can prove to the judge you have been supporting the child the she may be granted back child support. Next time she throws out that line tell her what her portion of rent will be from now on if she's going to play that game.