Are two separate rings in the same room (same phase, obviously)
"allowed" and are there any labelling requirements (apart from in the
CU) to distinguish them?

No it's fine. However, I would recommend mainly extending the existing ring
and using the new ring only for heavy fixed appliances, unless the old ring
is already heavily loaded, or already covers a substantial floor area.
However, the final decision will come down to how the individual appliances
are expected to be used. I fully expect to use tumble dryer, washing machine
and dishwasher simultaenously. Indeed, it is the normal case, so didn't want
to allow much diversity on that circuit. The tumble dryer is even capable of
running at 16A on its own, if I can get round to wiring it up safely.

I have three socket circuits in my kitchen. One is a conventional ring final
circuit on a 32A RCBO. Another 32A MCB (no RCD) radial powers only
integrated appliances (w/m, d/w, t/d) and has concealed sockets individually
switched by a grid switch above worktop level. The final circuit is a 16A
MCB radial purely and solely used for the fridge-freezer, again switched
above worktop level, (although by a 13A FCU rather than a DP switch for some
reason known only to myself).

Christian.