Temp agencies are a pretty nontrivial force in the employment world. If you work at a temp agency, they pay you regardless of whether you're bringing money, they do technical training and career advancement work, all the while you're "on deck" not actually being useful. Then a company like Google or Boeing or Joe's Game Studio comes along and says "we need 3 highly skilled C++ programmers in two weeks, for 6 months", and the temp agency sends you off to work there. You relocate, you work on some internal project for 6 months, and then you're back "on deck" at the temp agency, and then off to somewhere else. The company does not have the option to hire you permanently unless the temp agency put that into the agreement (probably for a price), the agency charged something like 2x or even 3x your salary to the company while you were working there, but your wage and benefits stay the same throughout, working or not.
So, who's your real employer?
Looking at this from an armchair, it's easy to miss that there's more going on than just, "Tim is on the team for the next few months." Whether or not you like the fact that this kind of agency exists and this kind of work is possible is another question.