I had a player come up to me and claim that he had heard from several judges that continuous deck building was allowed for a constructed (standard) a Regular REL FNM. I don't collect deck lists so it is not possible to strictly enforce this but I also told him that the expectation was to stay with the same 60 card MD and up to 15 card SB and that the deck needs to revert between rounds.

Am I missing something?

This option only exists at Limited tournaments. The relevant citation is MTR 7.3:

Players participating in Limited tournaments that do not use decklists may freely change the composition of their decks between matches by exchanging cards from their deck for cards in their sideboard without being required to return their deck to its original composition before their next match. The Head Judge or Tournament Organizer must inform players if this option is not being used prior to the start of deckbuilding. This option is not available at Competitive or Professional Rules Enforcement Level tournaments.

Continuous construction is only a thing at limited events run at Regular
REL without decklists (and where it wasn't otherwise disallowed), as per
MTR 7.3. There is no similar allowance in the section on sideboard use in
constructed tournaments (MTR 7.2)
The limited continuous construction rule is the only allowed exception to
MTR 3.15 which states: “The deck and sideboard must each be returned to
their original compositions before the first game of each match.”