This is even part of the FRCP - see 65(d)(2)
The only exception is basically aiding and abetting.
This is well settled caselaw basically forever. I posted it elsewhere in the thread, but here's a very recent example: https://reason.com/volokh/2025/01/24/google-as-non-party-not...
(Youtube can't be ordered by a federal court to take down posts as part of an injunction, when they aren't party to the case).
The court may want to keep whatever scheduling it has, but it's going to royally screw itself if it tries to order an injunction that affects the rights of non-parties in a meaningful way like this, without being really really careful about it.
Especially with SCOTUS just itching to find important cases where they can reinforce the limited nature of injunctions and ensure courts only grant relief that deals with the parties at hand.