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Strauss wrote a 3-act operetta called Hoch Oesterreich! It has the opus number 371.

He then wrote a march using motives from the operetta, also called Hoch Oesterreich! Usually this would put it on its own page, because it is not a straight extract of a whole number from the opera.

However, he gave it the opus number 371, same as the operetta.

Where should it go? I need to know whether I am tagging the march or the operetta . . .

No, Hail Austria is from the operetta Cagliostro in Vienna.
None of Strauss's operettas have opus numbers, but every one of them has concert music rearranged from motives (as you said) of which Op. 371 is but one. All of opera 369–374 (this includes the pretty well known At the Hunt, Op. 372) are from Cagliostro in Vienna, but they are independent works

So there's no issue here, at all.

OK, the Grove Music list must have confused me. So if I have this correct now, the operettas themselves have no opus numbers?

Right. All the ones called operetta have no opus number, only the dances arranged from them do (but NOT the few excerpts, such as the Entry March from The Gypsy Baron)

The only real confusing one is Ritter Pasman, which is a grand opera. I've seen the opera credited as 441, all of the excepts collectively 441, or JUST the ballet music as 441.

Melodia is correct. All of the operettas have multiple dance pieces (marches, polkas, waltzes) as spin-offs, which were assigned opus-zahls. The operettas themselves were not, nor were some of the more popular excepts from the operettas, like the overture to Fledermaus.

OK, they've pretty much all been tagged and linked to the correct operettas now.