Edited my original comment, I am a contractor on both, but on job #1 is mostly on paper. I am treated as a FTE.
I’ve had a similar situation before, and I will say that of course they want to make you feel like an FTE while keeping you on contract. That is the best of both worlds for them, of which you reap nothing but self-employment paperwork.
But since we're on the subject, I actually prefer working as a contractor on any occasion. I pay way less taxes, have control over my retirement money, and negotiate silly perks like gym/co-working memberships into a bigger salary. FTEs in my company make 25% less (net) and cost 30% more to the company.