What is the 3 dots icon for?

In Photos, my 3 dots at the top right of the Library view shows this:

In album view, I don't get the 3 dots.

In Edit view, I get this:

So, as JimmyCMPIT says, what you get depends on what you're doing. If you don't get options, then you don't have anything installed to use with it.

the icon usually shows other options you might want to have available for the task you're working on.

for instance if you open your drive from the desktop the windows that appears has the ...

from that ... there are options for new tab, get info, sort, etc. they are related to common tasks to using that window. Another application may have entirely different options.

does your Mac do this in any application you click the ... or one specific app?

Two screenshots-- both are made with my having clicked on the 3 dots.

The first lets you restrict your view without including shared albums.

The second is in Edit mode, and, for instance, sometimes I want to do some things to a picture, like burn and dodge, that you can't do with Photos, so I use another app I have called GraphicConverter. It jumps to that app, I mess with it, save, and it jumps back to Photos/Edit where I can decide to keep the result. Lots of people do this with Photoshop or other editors. So some people have a lot more choices there than I do.

To find out what happens, I use a picture I don't really need, and try things out.

I, too, am on Monterey. However, I was using a Library that is synced with iCloud, so it has the "Shared" thing. If I use a Library that is not connected to iCloud, then I get nothing, as you do. So it's contextual-- what it does depends on the context it's in.