They're on the same breaker, but so is all the lighting upstairs (labelled on breaker as well).
Is this an issue?
Therefore all those lights are on the same circuit. No, not an issue.
Why does same breaker mean continuous circuit
I am not sure continuous is the relevant word.
That is the definition of a circuit - all items on a particular breaker or fuse.
(I presume that this means neutral and live going through all lights).
Yes - the live and neutral from a particular breaker or fuse.
Couldn't one breaker control multiple circuits, or maybe that's just not done.
No. It is not possible. That would be one circuit.
If one of the lights in question were not on the same circuit then, after the alterations, the live to it would come from one breaker and the neutral would return to another breaker.
This is potentially dangerous.
Plus, if the lights were on different RCDs then they would both trip when either light was turned on.