I have a severe doubt that he's ten Hag signing. He doesn't match EtH other profile signings (or intend to sign), except Arnoutovic who most likely just a cheap filler player.
We
know he wasn't an ETH signing, in that there was a piece in The Athletic that made it clear that he was a club target ETH signed off on once it seemed his main target FDJ wasn't likely to be coming this summer.
But that's not an issue in itself, in that every top club has a dynamic whereby some signings are driven by the club's recruitment team rather than the manager. Salah at Liverpool being an immediate success story that comes to mind in that regard.
Whether Casemiro was a club target or an ETH target matters a lot less than how we ultimately made the decision to sign him. So hypothetically:
Scenario 1: ETH wants FDJ, can't get him. There are no real like for like alternatives available of sufficient quality. The recruitment team proposes Casemiro as an alternative. ETH thinks he can make Casemiro work by moving towards a different midfield set-up than the one he had in mind with FFJ, so he agrees to the signing on that basis.
Scenario 2: ETH wants FDJ, can't get him. There are alternative targets he'd like in a similar mold, but the club fail to secure them too. The recruitment team proposes Casemiro as an alternative. ETH isn't sure he can make Casemiro work as he's a wildly different type of player to what he actually wanted. But he's desperately in need of some midfield reinforcements, Casemiro is a big name player and the club persuade ETH that signing him is better than signing nobody, so he agrees to the signing on that basis.
In both cases you would describe Casemiro as a club signing rather than an ETH signing. But scenario one is a lot more likely to be productive than scenario two.