Cat Kaomoji
Cat faces made of text, with the parts named. A kaomoji becomes a cat through two things — a muzzle in the middle and a matched pair of whiskers on the outside — and everything else is variation.
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The muzzle is the character that matters
ω works as a small closed mouth in any face, but in a cat face it is doing something more specific. Its two humps are the upper lip either side of the philtrum, and the dip in the middle is where the nose meets it. That is a cat’s muzzle seen head-on. Put ω in a plain face and you get a mild expression; put it inside a whisker frame and the reading snaps to cat and stays there.
The alternatives are all chosen for the same silhouette. ‥ gives a blank two-dot muzzle. ｪ, small half-width katakana, is a snout, and it is shared with dogs and bears — which is exactly why cats need the frame around it. ᴥ is the muzzle again with the fur line drawn in.
The circumflex changes job
In (^_^) a circumflex is an eye folded shut. In (=^･ω･^=) the same character is an ear. Nothing about the character changed; it moved to the outer edge of the brackets and something else took the eye slot. This is worth knowing because it is an easy way to break a cat face: read the ^ as eyes, remove the ･ dots to tidy it up, and what is left is a smiling human face wearing whiskers.
Whiskers have to match in width
The frame is the pair of equals signs sitting outside the brackets, and it only works when both sides take the same amount of space. = is ASCII and occupies half a cell in a CJK font; ＝ is the full-width form and occupies a whole one. Both make a perfectly good cat — (=^･ω･^=) and (＝^･ｪ･^＝) are the same face at different gauges — but mixing them puts the head off centre, because one whisker is twice the length of the other. The same rule governs the ^ ears and the ･ against ・ choice for the eyes: pick a width and hold it across the whole face.
Faces that are not framed at all
A few of the faces above drop the brackets entirely. ᓚᘏᗢ is drawn from Canadian Aboriginal Syllabics and reads as a cat loafing with its tail out; ฅ^•ω•^ฅ replaces the brackets with paws, using a Thai letter whose shape has a claw in it. Both depend on the reader having seen the framed version first — there is no muzzle-and-whisker structure left to decode them from.
Ears built outward from the brackets rather than inward are the whole basis of the bunny faces, and the eye characters here come straight from the happy set. The kaomoji index has the shared vocabulary, and the Japanese cat page runs a longer set of the same animal, with more of the slit-pupil eyes than there is room for here.