Morphology is often "messy" in this way.
One big reason is probably because there was no need for a semantically distinct word presidence. Though it does not occur 100% of the time, there is a strong effect that is often referred to as blocking (coined in Aronoff 1976, defined here in Aronoff 2005):
Blocking: the process by which a potential word is prevented from occurring in a language because another form with the same meaning and function already exists.
(The mechanism for blocking depends on your linguistic framework — what is important is that the phenomenon exists, whether you call it "blocking" or something else.)
The words residence and residency actually mean different things. That one meaning took the form residence while the other took residency is almost certainly arbitrary, as the semantic distinction between -ce and -cy forms can't be generalized across words. But since each word had a semantic niche to fill, each word was able to become established in the lexicon. (And, with both words established, it might be possible for someone to occasionally substitute the word residence for residency or vice-versa, for whatever reason.)
As for presidence, what might it mean that is distinct from presidency? The fact that there is no answer is probably why such a word does not currently exist. It could, in theory, come into existence (existency? :) some day, should the need arise; there is nothing actually preventing its existence.
The last question that remains is, if only one word is needed then why should it have been presidency to begin with and not presidence? The answer is probably because of the differing origins of residence and presidency.
Residence entered into English directly from French. This -ence form is common in French; in fact, the word for presidency in French is presidence.
Presidency entered into English by way of post-classical Latin. So, we anglicized the Latin word praesidentia without letting the French get in the middle (as they often do) and throw away a bunch of sounds at the end of the word. So, praesidentia became English presidencie and later presidency.
Once they became part of the English lexicon, principles like blocking probably kept them organized as they currently are.