Not unseldom is in fact a stock phrase, predating Austen, and meaning "not rarely" or "not infrequently".
Indeed, the OED's entry for unseldom defines it solely in terms of this phrase:
not unseldom (misused for), not rarely, not infrequently.
It notes further, "Dutch niet onzelden is similarly used." and its citations range from before 1657 to 1882 (Pride and Prejudice was of course published in 1813, well within that range).
Of the various reasons we may have to use double-negation, I suspect the aim here was de-emphasis (i.e. "not unheard of" means something is indeed heard of, but de-emphasises that so as to suggest there may still be a rarity), that got muddled in an attempt to do so with a word that best fits through single-negation. That though is a matter of what happened with the word some 150 years or more before Austen.
In any case, by Austen's time it was an idiom best understood as having its own definition, rather than through analysing its components.
Not that Austen didn't use no double negation of her own. Austen's own double-negative constructions are generally of the sort of de-emphasising use for undertatement I suggest may have led to not unseldom coming into existence, and so it suits her well. She tends to use them with a certain gently mocking wryness, as again this example shows, so while a stock phrase rather than being Austen's construct, it is a phrase that suits her style well.
Ironically, understatement can have a nett effect of focusing our mind on something, and so while it de-emphasises on one level, it actually ephasises at a subconcious level. This is called litotes in rhetorical theory, and something Austen was adept at.