OK, thanks. It isn't immediately clear to me why it is picking up the conda in your external Anaconda installation, since `where conda` lists the Pro provided one as a priority. But it clearly is, since all the information you posted from `conda info` is from the Anaconda environment, nothing matches what we ship. Could you try removing the Anaconda environment from your path? It is known to cause problems to have multiple Python environments on the PATH, here's what the Anaconda installation says:
The other option is, if you don't really care which conda executable you're using, is to specify the full path to the environment you want to modify. So, do something like:
activate "C:\Program Files\ArcGIS\bin\Python\envs\arcgispro-py3"
conda upgrade -c esri --no-pin arcgis
It looks like you have two environments called "arcgispro-py3" -- the one that ships with Pro, and a second one located in F:\Anaconda\envs. That likely is confusing this arrangement, since the message may be perfectly valid (if there is no arcgis package in that separate "arcgispro-py3" environment).
Hope that helps, and happy to keep digging if this doesn't get you where you need to be.
Cheers, Shaun