Short answer: Fossils are version-exclusive.
Long(er) answer: Some fossils are version-exclusives, but the Pokémon that are restored from them are not formally considered version-exclusives.
Prior to Generation IV, Fossils were Key Items. However, from Generation IV and on, they are normal items, meaning that they can be traded freely between games.
Essentially, this means that before DPP, fossils couldn't be traded, only the Pokémon could, which made the Pokémon version-exclusive. Now, since the fossils can be traded, all fossils can be restored in any game, therefore you can technically obtain any Pokémon from a fossil restoration location.
Now, in USUM, obtaining fossils themselves is version-exclusive, with the following fossils in each game:
Ultra Sun:
- Helix (Omanyte)
- Claw (Anorith)
- Jaw (Tyrunt)
- Skull (Cranidos)
- Cover (Tirtouga)
Ultra Moon:
- Dome (Kabuto)
- Root (Lileep)
- Sail (Amaura)
- Plume (Archen)
- Armor (Shieldon)
The Old Amber is unobtainable in USUM, but Aerodactyl is available for in-game trade in Seafolk Village. Also, these ten fossils are available regardless of version in Poké Pelago. Finally, these ten fossils can be purchased at Olivia's jewelry shop in Konikoni City.
Hope I helped!
Bulbapedia's page on Fossils