In quantum mechanics? Energy conservation is not as strict as you might think.

It just somehow always balances out in the end.

I know, there's some tricky business with energy conservation (I'm not pretending I understand it all though). It is for me an easy analogy though. Even Hilbert space has to stick to some rules!

Yes, and in a more general way Noether thm applies in QM/QFT as well (energy conservation is just a special case). Symmetries become operators acting on the Hilbert/Fock space (ie representation).

With the caveats: - anomalies-> symmetries/conservation broken by quantum noise (has to be cancelled in valid theories) - Heisenberg’s incertitude

Aren't the laws of physics based on repeated observations that continue to show, for example, that we always see energy being conserved?

I'm not sure we can extrapolate that out and say that can never be violated, only that we've never seen it happen.

Energy is not always conserved. In the universe as a whole, energy is not conserved because we do not have time symmetry.

Sure, that could be a reason why it isn't conserved. My point wasn't that we know ways it isn't conserved, only that theoretical understanding isn't universal or necessarily true.