I would imagine this might be a bit tricky as you would have to write a routine that analysed each game and identify it from an opening book database. I can see some merit in the idea but I think it would be very time cosuming to code and may well put a strain on the speed of the games database.
Difficult to implement, yes. Compute intensive - not necessariiy! There are some very efficient chess database programs out there, most of which are able to identify a chess opening very quickly.
Once the game's moves have been turned into text string in a standard notation, a simple search is all that's needed to find the opening whose representation in the same format is the longest exact match for the first part of the game. Whatever representation this site uses for a game in its database, it should not be too tricky to convert a reasonable-sized database of openings to the same format.
I had envisaged storing the standard chess opening code (D01 or whatever) as part of the game record. That way there is a small storage cost, but very small processing cost (similar to computing the FEN, less than checking for legality, I would imagine.) The opening code could be displayed in games lists at a modest bandwidth cost.
I am not sure whether using the FEN or the move list is better. After a few moves, the FENs would no longer appear in a list of openings, but all move lists must have a leading string that will match. On the other hand, using the move list would fail for unusual transpositions. Whenever you have a FEN that doesn't match or a move list that is not an exact match, you stick with the previously computed opening code, so maybe it's down to the size of the representation of an opening: a move list will be more compact than a FEN for the first few moves.