For example, say you have 2 levels in Dweomerkeeper, meaning you have Arcane Sight (Su). Does this count as one of the two required divination spells to take levels in Unseen Seer?
Complete Arcane gives characters the opportunity to meet some spellcasting prerequisites with spell-like abilities, but
This explicitly does not cover “ability to cast Xth-level spells” as found in unseen seer’s in requirements. The qualification on those spells requiring some to be divinations doesn’t change that.
This also doesn’t cover supernatural abilities anyway.
Unlike spell-like abilities, which almost-always copy the effects of some specific, named spell, supernatural abilities rarely have anything to do with spells. The dweormerkeeper’s ability is unique in the rules (and also horribly broken, please don’t actually use dweormerkeeper in an actual game, at least not without nerfing).
The only other class with a significant suite of supernatural abilities is the Tome of Magic binder, whose chapter also details a number of “meta-supernatural” feats to improve those. Tome of Magic makes no mention of the binder ever qualifying as a spellcaster, and it would be decidedly weird for it to do so, considering the existing of the anima mage and Tenebrous apostate prestige classes which combine binding with spellcasting. And we know that Tome of Magic’s authors were aware of the possibility of the new magics “counting as” spellcasting, because the following chapter about the shadowcaster has it doing precisely that.
So all told, the answer is, officially, no. You can ask your DM for a favorable houserule, which I’d generally be inclined to offer, personally, but not for dweormerkeeper. Well, that wouldn’t actually matter because I wouldn’t allow dweormerkeeper in the first place.