The fairphone 5 official ROM is bloated with about all the google app you can imagine. While it is possible to use it without google account, you will need one to install applications from the official google play store.

You can still use an Alternative store, like the aurora store (download it with f-droid) that can connect to the play store with anonymous token.

If you consider not using google account, I recommand you to install one of the following :

• /e/OS, that has an ad/tracker blocker (that uses the VPN spot), micro G installed and configured (open source version of google framework) and a store that more or less is a fusion between aurora store and fdroid.
• iodéOS : it has an ad/tracker blocker (that uses the VPN spot ) + microG configured.
• CalyxOS, that will let you choose if you want micro G or not. It comes with the aurora store and fdroid installed.
• lineageOS, you can choose if you want to have the google apps, micro G or nothing depending on the version you download.

If you choose an alternative ROM, you'll probably want to download a Gcam app to improve the pictures quality.