He’s not my “keeper” in any way, shape or form, and I don’t want him to be, either.
That would be “keeper” as in “zookeeper”.
In this context ‘keeper’ is code for ‘ruler’. Now it all makes sense doesn’t it. BTW it’s hard to fathom what the heck this speech was really about. It certainly looks like a tattle-tail whining about no getting his own way. Pathetic for sure....