I always see kohanim remove their shoes before n'si'as kapayim, even those without leather, but not their socks. In other contexts I'm familiar with, socks are the same as non-leather shoes. For example, we can wear socks or non-leather shoes on the ninth of Av, and (I think) a kohen while working in the bes hamikdash can have no separation between himself and the floor, thus neither socks nor non-leather shoes.
Are the kohanim I've seen correct to remove even non-leather shoes but not socks?
what's the difference between socks and non-leather shoes? I mean: What's the definition of what must be removed, which includes even non-leather shoes but not socks?
The reason kohanim remove shoes before nesiat kapayim is due to a concern that a strap may break while the kohen is going up, causing him to stop to fix it and thus miss nesiat kapayim (Sotah 40a). Aruch HaShulchan 128:12 rules that non-leather footwear with the same type of straps would also be forbidden, for the same reason. (It’s not clear to me whether non-leather shoes with modern shoelaces would fall under this rule.)
Even absent the issue of the straps, Mishnah Berurah 128:18 (quoting Peri Chadash) rules that non-leather footwear which is worn outside in the mud should also be removed, due to considerations of respect for the congregation.
So there are (at least) two possible reasons to distinguish between socks and non-leather shoes when it comes to nesiat kapayim: the type of fastening on the shoe (if relevant) and whether it’s worn outside.
Where neither of these issues arise (e.g. if wearing non-leather bedroom slippers on Yom Kippur) I was taught by my father (he should live and be well) that a kohen need not remove them for nesiat kapayim, analogous to socks.