Subsystem codes have observables that commute with all measured operators. The honeycomb code as normally described has measured operators that anticommute with any possible choice of observable. It bypasses this problem by using dynamical observables. They always anticommute with some of the measured operators, but never the ones you just measured or the ones you're about to measure next. So the honeycomb code is not a subsystem code.
Here's what you get if you take make a detector slice diagram of the honeycomb code circuit, as it executes. I made this diagram by downloading the
circuits.zip file from https://zenodo.org/record/7072889 and running
stim diagram \
--type detslice-svg \
--tick 0:46 \
--out tmp.svg \
--filter_coords "L0:*" \
Each of the squares is the checked stabilizers at some time (the stabilizers that will be measured in order to produce detection events). I've picked a size that makes them repeat in line with the grid. Focusing on just one of those columns we see this repeating thing:
It's hard to tell, because some of these things are overlapping, but the bright red rectangles are each a checked XXXXXX stabilizers, the bright blue rectangles are each a checked ZZZZZZ stabilizers, and the muddy red operators are each a pair of overlapping checked XXXXXX and ZZZZZZ stabilizers. The small blue circles are showing the vertical observable; each blue circle is one of its Z components.
Now forget about the honeycomb code and just look at this picture. This is a picture of a stabilizer code. It has a set of stabilizers to measure, and an observable to protect. It has a code distance. It's actually sort of similar to a color code, but with 1/3 of the stabilizers missing.
The circuit that I loaded to make this picture can be thought of as measuring these stabilizers, while protecting this observable. Sure, there's some shenanigans that occur where there's feedback into the observable from the measurements, but that kind of intermediate complication always happens when you turn codes into circuits. If it wasn't measurement feedback it would be perturbations from CNOTs.
So the honeycomb code is not a stabilizer code, or a subsystem code. But when you compile it into a circuit all that context is forgotten. And if you took the stabilizer code I drew above, and compiled it into a circuit, you could legitimately get the same exact circuit that the honeycomb code uses. It might not be the first circuit that would occur to you, but it's a valid one. So, ultimately, the question is kind of meaningless, because once these things are translated into circuits it all kinda mushes together.