Mouse won't left click anymore on iMAC

Is it setup as a left handed mouse?

Reinstallation would set up a default mouse right? Meaning it would left click.

It doesn't left click even during installation.

Does the mouse work with a right click?  The default Mac mouse is actually a 2 button mouse.  The right click is initially set to function the same as a left click, making the mouse functionally a one button mouse until you change the settings in the mouse preferences.  If that doesn't work, try a different keyboard.  Maybe one of the control keys is stuck.  On a Mac the control key forces a right click for a 'one button' mouse.

Please guys, read the question. This happens with any mouse. I have tried 3, one of them was an Apple mouse, 2 were not.
Left click doesn't work with any of them, not even in the OS setup interface.
Right click works only in the OS setup interface. But I'm not keen on reinstalling, since it seems to have deeper problems.

What model is it? If it has a wired keyboard, disconnect it, if it's wireless, remove batteries. This is to be sure no keys are stuck.

Yes I removed the batteries and switched off the wireless mouse.
Now I am using a generic Wired keyboard and a Wired mouse but the problem is still there.

I managed to reinstall because the left click was mysteriously working on one of my tries. It kept on working after setup for as long as I kept the iMac turned on. After I turned it off and on again, I lost the left click again, both with the standard mouse and with other mouses.
Since I disabled password logon, I was able to see something: it looks like the left button gets stuck, right away of after a minute. I noticed this because I was moving the mouse on the desktop without clicking and it started drawing a rectangle with a fixed vertex. After this I was unable to click once more.

So after a reboot you have some time the mouse works? During that time, can you disconnect all but your wired mouse? See if it stays workign then? I never saw something like this before btw, I'm suspecting a hardware issue (since you reinstalled).

I removed the batteries from both the mouse and the keyboard, but I have the same problem with wired mouse and keyboard. Yes it's mind boggling.

I understand but what I mean is disconnect the wired keyboard once the wired mouse is working. If the mouse then stops working (again) it's likely a hardware issue.

Ok, I found the problem and it's HILARIOUS. These customers are actually my neighbours and their home is perhaps 20 mt from my lab.
They had at home an Apple trackpad turned on and connected to the Mac all the time it was in my lab. Bluetooth can travel quite a long distance, so the Mac still "felt" that trackpad and somehow the trackpad conflicts with any mouse. When it's on, any left button won't work.

I had a flash this morning and asked the customer "Do you by any chance have another Apple mouse turned on at home?". Bingo.

I think it's one of the most absurd things I've seen since I work in this field, and there were MANY.

ROFL - I should have asked you to look for Bluetooth devices 😀😀😀

Glad to hear that you found it!

What's funniest is it was in their home. All good now though. They get a clean install with all data recovered, so I think they'll be happy nonetheless.

I guess I should click "I found my own solution"?

Yes, you've found your own solution, helpful comments can be selected but is not required  ;)

When I was testing devices, I've had multiple bluetooth mice and trackpad attached without issues.  I think it's because the device was just at the limit of its range that was causing interference, or they had something resting on it that caused a capacitive touch trigger.

Actually they had the same problem at home. I just told them to keep the trackpad turned off, since they never use it.

It sounds like something's wrong with that trackpad.  I've never had that kind of interference with multiple bluetooth mice/trackpads.