I assume you’re referring to the current model (FP3). The bootload can be unlocked. So I’d say it’s not restricting anything. But I think it’s anyway not possible to install Linux on it.
Maybe it’s a good start for you to have a look at the oslist.
Fairphone ships Android without much bloatware, in any case. On the FP2 the Google apps are preinstalled, as is the Fairphone updater (to update the OS) and iFixit (which is tiny.) I don’t have an FP3 at hand to see which apps it contains, but it doesn’t have much more.
And both have an OS without Google apps if those bother you: Fairphone Open for the FP2 and /e/ for the FP3.
The Fairphone OS on the FP3 is pretty close to stock Android. It has the usual Google services and apps installed, but no bloatware. Fairphone has made some modifications, for example their (proprietary) smart charging, which does not use Quick Charge for the last 10 percent of the battery capacity.
As of now, there is only one official alternative OS available for the FP3, namely /e/OS, which is a fork of LineageOS. Officially still in beta, btw.
The Linux distros which are currently available for phones are at best proofs of concept, but not suitable for average and every day use IMHO. More importantly, they’re lacking the rich app ecosystem that Android provides.