“Catercorner” (also written “cater-corner”) is one of those words that spawns a lot of variants. It originated in the mid-19th century from the word “cater”, meaning the four on dice, derived in turn from the French “quatre”. It means “diagonally opposite”, and is quite a useful word.
It is pronounced “catty corner”, and that pronunciation has led to the mistaken association with the animal, and that of course has led to the variant “kitty corner”. In fact, if you look up “catercorner” on merrian-webster.com, it defines “catercorner”/"cater-corner" as a variant of “kitty-corner”, even though it then goes on to give the derivation as “alteration of cater-corner, from obsolete cater four + corner”.
The English language (and probably others as well), has a number of consonant sounds that are voiced and voiceless cognates: that is, sounds that are produced the same except that one of the pair is made using the vocal chords and the other is not. The sounds of “t” and “d” are cognates. Almost all speakers, at least here in the U.S., replace each of those sounds in the middle of a word with something called an “r-tap”. (Why it is called that, I do not know.) The word pairs “latter”/“ladder”, “mettle”/“meddle”, “matter”/ "madder” tend to sound alike and need to be distinguished by context. It’s quite common for phonetic spellers, therefore, not to know whether to spell an unfamiliar word with a “d” or a “t”. (Think of how often you see “congradulations” written for “congratulations”, or “congrads” for “congrats”.)
So we not only have “catty corner” and its sister “kitty corner”; we also have “caddy corner” and “kiddy corner”. Which one you say or write often has to do with where you grew up, but it could just depend on which variant your family uses. I grew up with “catty corner” and write “catercorner”, but a neighbor and close friend of my mom’s used “kitty corner”. They each grew up in different places.
So if you were wondering which version is correct, apparently one can use “catty-corner” or “kitty-corner” in place of “cater-corner” with propriety, but leave the caddies and the kiddies out of it, unless the caddy is standing catercorner from your kiddy on the putting green, yelling at him not to touch the ball.