Chrome installer running every time I open Chrome

If there is an "Applications" folder beside the Chrome app within the Finder window of the mounted DMG file you downloaded, then drag the Chrome app onto that Application folder beside it. You should be prompted for a password to allow the transfer.

If you see a small curved arrow when dragging the Chrome app, then you are creating an Alias which is wrong and likely means you have some stuck keys or some odd configuration issue. If that happens try holding the Option key down while you drag the Chrome App to the Applications folder.....you should see a green circle with a plus sign indicating you are copying the item. If you don't get the green circle with plus sign while holding the Option key while dragging the file, then have some sort of configuration issues with your user account or some other software issues.

If none of this works, then try creating a new admin user account. Log out of your main user account, then log into the new admin user account. Try dragging the Chrome app to the Applications folder. If this works, then you have confirmed an issue with your main user account. If you still have this issue with a new admin user account, then you have a system wide issue of some sort.

From Google Chrome for the Installation on macOS

When you Dragged the Google Chrome Application into the Applications folder

Did it Ask for the Admin Password ?

If Yes, did you enter the Admin Password and therefore, Given Permission to be Installed in Applications folder ?

User wrote " EVERY time I open Chrome, the installer runs and I have to drag the app to my applications folder, creating a shortcut."

In Apple Parlance - a Shortcut is an Alias that Points to the Actual Application and its' location

User wrote " Nope, it does not ask for the admin password "

That in and of itself, would basically confirm that a Alias was created which would Not Present the need for the Admin Password

Something is wrong if dragging to the Applications folder creates a shortcut (alias) instead of actually copying the application. I don't use Chrome and will not install it on any of my macs, but if you want to, make sure to follow the installation instructions.

Related or not related

" I have an M1 MBP that got a new logic board/SSD last month. I did a restore from a backup "

Personally, I may restore Only the User Account and nothing more

"  Issues that may exist on the TM Backup when it was performed, will be migrated to the New Installation "

This would badly pollute the new Original and Pristine installation of the new OS.

Whether or not from a Time Machine Backup or some other way or means.

Since the user brought it up " I did a restore from a backup " everything is relevant and fair game for trouble shooting this issue

Had it not been mentioned anywhere, then Yes, it would be Off Topic.

You and I have gone some what off the original question

Sorry about that - really

In which case I leave you in the very capable hands of my good colleague @ Luis Sequeira1