Are Arpeggios and fingerpicking patterns the same thing? Is it simply playing the root, third, and fifth over and over for a given chord?
Arpeggio is the instrument non-specific term. To play the tones of a chord separately instead of simultaneously.
Fingerpicking is specific to guitar (and other string instruments I suppose) about how to perform the music. Although fingerpicking doesn't necessarily mean arpeggios. You can play full chords with your fingers.
Also, some may consider an arpeggio to play the chord tones in ascending order (sometimes descending) as apposed to some alternate picking of the tones like in Travis picking.
Fingerpicking on a guitar could mean several things. It could be fingering a chord shape and playing strings in a pattern. There are many different finger picking patterns most of which use the notes from chords, but they're not strictly going to be arpeggios. They may or may not constitute arpeggios - but could come close. It could also mean simply using fingers rather than a pick, and that could be a melody, which would include notes that are not in a particular arpeggio.
But - to play an arpeggio properly, one would play each and every note in that arpeggio is strict order, bottom to top, or top to bottom.So, for example, a C major arpeggio would be C E G C, or C E G C E G C. Both of which are playable on guitar.
You could play arpeggios on guitar with a pick, so that wouldn't be fingerpicking per se.
So, they are not the same as each other, but there is a small common area where they could meet.