If you wanted fdroid to update apps automatically ~~you'd need to have some system level service running like Google Play services. This is course could be achieved with a custom rom or using root. (This is how huwai and other devices that don't get the play store, as well as Samsung handle their own stores)~~ [untrue as of Android 12, see below comment]

That said I view automatic updates as an anti feature most of the time. I should be asked if I want updates. You can of course turn off auto updates in the play store too though so that's more of a side note.