First of all, the following info only applies to a Steam installation of MSFS.  I don't know the file locations for a MS Store installation.

After installing MSFS via Steam, the "Options" and "Personalized Controls" profiles are stored in the following location:
...\Steam\userdata\[your user number]\1250410\remote

The "Options" file is named "Profile_00" and is a binary file. Deleting it will cause MSFS to rebuild it from scratch on the next startup.
You could build several profiles by renaming them after being built.  Then just copy the one you want to "Profile_00" before the next startup.

Any personalized "Controller" profiles are stored in "inputprofile_xxxxxxxxxx" files.  They are XML files which can be edited using a text editor like notepad.
In each file, near the top, will be the name you gave it when you created it in MSFS.
For reference, the "Default" controller files are located in:
...\Steam\steamapps\common\MicrosoftFlightSimulator\Input

Also located in the "...\Steam\userdata\[your user number]\1250410\remote" directory are two files named "kh_logbook" and "datausage" which are binary.

Thanks for this, moved to a new computer and what a PITA to move joystick configs.  You can't just copy the inputprofile files into your steam folders and have them be recognized. Thank god they're text files, I was able to create a new profile with one entry and then edit that profile coping the text from the old profile to the new and that worked.

Thanks so much for sharing this!

Wonder why they chose this way - certainly not the most user friendly approach. I wish it was as intuitive and maintenance & sharing friendly as in DCS.

Thanks, Important stuf for those doing custom profiles.