advowson: If you have not yet, I suggest you download Jarulf's Guide and read the relevant sections. It is widely considered to be the definitive reference for the game, easily surpassing official publications in detail and correctness. The most pertinent to your question seems to be section 3.8.3, page 42, which says in part:
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If several unique items are possible, the one with the highest qlvl will be dropped. If several unique items that are possible have the same qlvl, the one that happens to be first in the list will be dropped. This is the reason that some unique items will never be dropped as there is always another unique item with the same qlvl (and being of the same base item) that will be dropped instead. See chapter 3.5.2 for a list of such unique items.
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The rule about "highest qlvl" is why some items are very hard to get: there are few sources that are high enough level to drop that item, yet low enough level not to drop some other, higher quality, item. For example, Sharp Beak is a qlvl 2 Large Axe (3.5.1, page 30) and Wicked Axe is a qlvl 5 Large Axe. To get Sharp Beak, you need a monster to drop a unique Large Axe and for that monster to be unable to drop a qlvl 5 (or higher) unique item. If it
can drop a qlvl 5 unique item, it will drop Wicked Axe instead of Sharp Beak. Per 3.8.2 page 41, the base item is in the range 1 to mlvl(battle). According to section 3.8.3, page 42, unique items dropped by unique monsters have a qlvl in the range 1 to ilvl+4. According to section 3.1, page 17, a Large Axe has a qlvl of 6. Therefore, you need a monster that has a base mlvl too low to drop Wicked Axe (so, mlvl 4 or less) and an mlvl(battle) (mlvl adjusted for difficulty) high enough to drop a Large Axe (so modified mlvl of at least 6). This is why you cannot get Sharp Beak on Normal difficulty, but you can on Nightmare or Hell difficulty. Even then, you can only get it from low end monsters, because higher level monsters will drop Wicked Axe instead.
From this, it's not purely about the number of drops, but also the quality level of those drops.
I had a read of those pages... goddamn. Estimated 2900 times clearing level 1 for a chance to get a Rift Bow... Just. Wow.
Welp, here I go trying to get a legit Ring of Regha, Bleeder and Bramble!
EDIT:
Welp, after two hours of playing floors 1 and 2 on Nightmare, I got The Celestial Axe and Gladiator's Bane, as my two new uniques. I also got Constricting Ring, Torn Flesh of Souls (on the floor of a treasure room!), Bow of the Dead, but I already have those ones.
When they say you need multiple uniques of the same qlvl to get certain rarer ones, they have to drop in that game right? I can't bring a Ring of Regha to a new game to force the game to give me a Bramble?
EDIT 2:
Okay, I think I answered my own question. No, bringing a unique item to a new game does not force the game to choose a different unique I think. I just got another Gladiator's Bane. And a Black Razor (new!).