The issue is indeed an incompatibility between the version of libstdc++ shipped with Julia and the (newer) one expected by matplotlib.
The compat bound libstdcxx-ng = ">=3.4,<14.0" that you see in the CondaPkg.resolve() output is supposed to solve this, by restricting Conda to only expect versions older than what is in Julia (because older versions will be ABI compatible with what is loaded).
However the error you are seeing suggests that actually libstdc++ 14.1 is expected, which is outside of this range.
Conda has installed libstdcxx-ng 13.2 but libstdcxx 15.1, which I suspect is the problem. CondaPkg doesn’t restrict the latter.
Can you try running the following and then see if matplotlib works?
CondaPkg.add("libstdcxx", version="<14.0")