Battle of Hoover Dam

As this enchantment enters, choose NCR or Legion.
• NCR — At the beginning of your end step, return target creature card with mana value 3 or less from your graveyard to the battlefield with a finality counter on it.
• Legion — Whenever a creature you control dies, put two +1/+1 counters on target creature you control.

Fallout

As Battle of Hoover Dam enters the battlefield, choose NCR or Legion.
• NCR — At the beginning of your end step, return target creature card with mana value 3 or less from your graveyard to the battlefield with a finality counter on it.
• Legion — Whenever a creature you control dies, put two +1/+1 counters on target creature you control.

Finality counters aren’t keyword counters, and a finality counter doesn’t give any abilities to the permanent it’s on. If that permanent loses its abilities and then would go to a graveyard, it will still be exiled instead.

Finality counters don’t stop permanents from going to zones other than the graveyard from the battlefield. For example, if a permanent with a finality counter on it would be put into its owner’s hand from the battlefield, it does so normally.

Finality counters work on any permanent, not only creatures. If a permanent with a finality counter on it would be put into a graveyard from the battlefield, exile it instead.

If you somehow control Battle of Hoover Dam and no choice was made for it (perhaps because another permanent on the battlefield became a copy of it), it has neither of the two triggered abilities.

Multiple finality counters on a single permanent are redundant.