I'm on Win10/64 bit.

So, this report is for Blender version released on 7/8/2019. I do not have the 7/8 version. I tested with the 7/7/2019 version and could not replicate. I downloaded today's version (7/9/2019) and tried again. This time I was able to reproduce the reported issue in the uploaded file.

I think the issue has something to do with the subdivision surface modifier, but not wholly.

If I have the modifier showing in edit mode but have the 'On Cage' setting disabled, I cannot select from the upper skull model in face select mode. If I turn on the On Cage setting in the modifier, I can select in face mode in edit. Or if I disable the modifier in edit then I can select faces just fine.

However: I tried to replicate this in the default file by adding a sub-surf modifier to the default cube, ensuring the modifier was active in edit mode, and On Cage was disabled. This time, I was able to select faces in edit mode just fine with the On Cage disabled.

So I don't think it's as simple as a broken subsurf modifier.

So to sum up: edit/face selection for the uploaded file worked in 7/7 build, but not in subsequent builds. Tweaking settings in the subsurf modifer can restore face select functionality if On Cage is enabled. However, the subsurf modifier doesn't create the same issue in the default cube with identical settings.

There weren't any console errors that I could see.

Hope the additional information helps the devs.

Edit: note there are some empties with missing images in the blend file. Make sure you pack all assets into any blend files you upload.

This looks like a duplicate of T66630