If you installed the browser from the Software Center, you need to know there are two possibilities:
The one named "Chromium" is what is known as a snap version and you can read more about snaps here.
Notice the source is the Snap Store and the download size is 141.6 MB.
The one listed as "Chromium Web Browser" is the regular version, like most of your other software, from the repositories.
The source is the universe repository and its size is only 57 MB.
(By convention, if both a snap and a regular version are available for the same application, the snap version is listed above the regular version. See here.)
Your browser profile in the case of the second version, the regular version, will be in
~/.config/chromium as you pointed out in your question.
With the snap version, things are different. Snaps you install as a user will be in your home folder in the
To find your profile folder for the snap version, point your file manager or terminal to
Given the nature of snaps, you may find your experience a bit restricted in interacting with the rest of your system.
If you have indeed installed the snap version and wish to remove it, make sure it isn't running, open a terminal and execute
snap remove chromium. You'll get an interface asking you for your password. Provide it and press Enter. Done. Keep in mind that everything in
~/snap/chromium, including your profile data, will be removed!