I just created a *.provisionprofile through the Apple developer portal and I am not sure if I can add this to my Git repo. The documentation refers to this at multiple locations, but I am unable to find further information if this file is something that can be shared.
Is this a file that contains private information?
Private, at least to the general public. You'd need to deploy it in your team's CI so your team would be able to see it too, but there's no risk of them getting access to your account with it. There's also the risk of others using it to generate a malicious copy of your app, but actually deploying it to a device requires more workaround. They can't just push it to App Store without breaking into your account, at which point they could just generate a new profile.