Perhaps anything holy would burn vampires but as crosses are easily recognisable, it makes sense to use them?
My headcanon is that lost in the ancient past (and perhaps by the same people that created the First Slayer) was an epic spell that affected all vampires (just as a spell can affect all Slayers, at least when one has the Scythe). Just as the Scythe was used for Slayers, I'd say the cross was used for vampires, because the cross would represent how a vampire is both demon and human (and some demons would say a vampire is not either, at least not completely). That is just as the cross is made of 2 intersecting sticks, so is a vampire the "cross" of human and demon. It's the symbol of two natures that is powerful here.
Either as a side effect or deliberate intent, the cross gained a mystical power against vampires, but the reason for this is magickal, not religious. The cross became a Christian symbol because Jesus died upon one, as have countless others (and the Romans weren't the only one to practice that form of execution), which makes it believable to me that the cross becoming a Christian symbol was just coincidental to the cross also being used in a magic ritual. Naturally, when crosses were seen to have worked then Christians would (wrongly) attribute the power of God rather than the power of magick...and even vampires raised in Christian lore could make that mistake.
That said, it begs why holy water work. It's not stated that ONLY Christian holy water works, but assuming that is the case then that would suggest another epic spell by one who is Christian, or at least aided the Church in exchange for helping wipe out vampires.
'Course another explanation is that Christianity is the one true religion (which strikes me as even sillier than the vampire lore of Buffy), but that opens another can of worms...like why are vampires the only demons affected by crosses, as well as having other weaknesses limited only to them?
And just for the record, I like how Stephen King handled such matter in Salem's Lot in which it wasn't the cross or anything else, it was the faith. The priest in it was called "shaman" and in the novel there are references to many forms of "white magic" that isn't restricted to Christianity that the vampire has come to grudgingly respect (if in a hateful way), though not the priest who is too much of an alcoholic to have the necessary faith. When the priest refuses to throw away the cross, showing his faith was in the trappings than in God, then the cross loses its power, something that disappointed the master vampire who had hoped he had found a worthy adversary. In contrast, a little boy with perfect faith is even able to ward off a vampire with a toy cross taken from a monster model set, while a couple of others thinking fast make a cross out of tongue depressors (saying Psalms 23 over it, IIRC). But in this version then the Star of David should also work since the trappings are irrelevant, only the faith is important.
