Myahele has the right of it. To extrapolate a bit, here's the long story short:

Is it akin to Louisoix's staff?
Visually, yes. Materially, not quite. It seems that this staff aesthetic has a varied and long history in Eorzea.

What are the marks on them?
The symbols of Wind aspected deities from among the Twelve.

Why do the Ixal have them?
They stole from Conjurers from among the Five Races (hamlet / caravan raids)

Doesn't that bother Garuda?
Apparently not (We asked Fernehalwes, just in case). As long as she gets her worship and crystals, ain't nobody got time for that.

Remaining Question:
Is there anything more to it, or did the team just feel that those Ixal looked best with those staffs? A few times in the past, people have wondered if there's any small hint in that wind-loving beastmen go out of their way to steal staffs with wind-aspected gods' symbols on them to cast their wind spells with despite worshiping the Lady of the Vortex and (seemingly) having all the wind power they need from that. Do the symbols have a deeper connection to the elements than has been revealed so far? Or are the staffs just constructed using wind-aspected materials that aren't visible over the decorations? More likely, are these people just digging too deep into it?