Just to clarify some common misconceptions …
Fairphone OS … is Android.
With Google Apps and services preinstalled, certified by Google, supported by Fairphone, currently Android 7.1.2.
Fairphone Open OS … is Android.
Without Google Apps and services preinstalled, supported by Fairphone, currently Android 7.1.2.
LineageOS … is Android.
Without Google Apps and services preinstalled, supported by the LineageOS community, currently Android 9.0.
/e/ … is Android.
Based on LineageOS currently, supported by the /e/ community.
Just different flavours based on the same Android groundwork by Google.
You can run some real non-Android OSes on the Fairphone 2, see #oslist.
Not by default, but you can install Google Apps and services yourself to your liking (you can choose exactly which ones) via projects like https://opengapps.org
And you can register a device running an Android not certified by Google with Google to avoid running into potential trouble using the Play Store on such an Android … https://www.lineageos.org/Google-Play-Certification
So if you really wanted, you could have your LineageOS outfitted with the same Google Apps and services like Fairphone OS … well, newer versions of them currently .
Difference is … if something is broken and can be traced back to Google’s groundwork, Fairphone can complain and demand a fix from Google (in my understanding at least) … the LineageOS community can’t do that, they would have to somehow fix it themselves.