[Clockworkers Doll]

In all seriousness, I've never had to deal with this so I can't relate and give you a more long post, but here we go.
Yes, it probably would've been better to only whisper it, pull you off to the side, or paging you (seeing as I think GMs can see your key, may be wrong but I think this is a thing from what I've seen) would've been a generally better idea. I've seen people who accidently break lore or get pulled over for stuff like this happen in OOC and IC and it brings up a lot of room for other players to shit on you or spread rumors and such. Some people will downright avoid you and RPing with you any way they can from this kinda thing. It's also just... Well, it's embarrassing for the player. Can't tell you how huge the heart-attack I had when I self-partied so I could move to alts and then have Chaos yell at me on OOC via the GM channel.
But that doesn't mean Sly did anything extremely wrong, and Blues is actually right so I won't go into it.
From experience, a good idea be putting disclaimers in an obvious, easy-to-see spot in your descriptions to avoid certain things like this. (A simple "(Disclaimer: Not really X/X is fake/etc.)" usually works.) Because I tried to make a character with costumed legs and then Sly got a bunch of comments because I didn't do that. Then I fixed it and shit chilled. I'm assuming you didn't have something like this if this had to happen. If you do, well, make it bigger. It saves the trouble, and if someone metagames that? Well. You just tell a GM about it and move on. And, uh, I'll go over two things that actually caught my eye when reading this.

So when someone I don't know comes out of nowhere and tells me that I can't play a stuffed animal, that doesn't register as 'Oh, this is an admin inquiring about how my character might be breaking the rules', it came off as a random asshole breaking up IC with antagonizing OOC when they didn't have the whole picture and just had 'people complaining to them' over skype.

This is kinda an issue too, and is completely understandable. Coming up within IC parameters can be confusing because both Sly and Chaos don't have things that can identify them as GMs. The cons are exactly what you were talking about, but the pros are that it's a bit easier to catch rulebreakers in the act when nobody knows that's a GM's character. Although most people who've been around for awhile know Reserved Thaumaturge is Slydria and Heavy-Layered Warlord is Chaos.

Skeletons with a Sans/Papyrus playby despite there being no skeleton race and the closest thing to that (Liches) still not fitting. Mechanations that go insane, can get pregnant, or otherwise experience everything that non-artificial life experiences. Players with faceicons of real life people or giant baby faces or what have you that obviously don't fit their character or the ingame universe.

Well, first... Liches have a skeleton skin. Therefore, they are allowed to have skeleton playbys. Case closed. Nobody complains about people that don't have Undertale skeleton playbys either, including myself who before and after Undertale was released, had a Lich who was legit just a skeleton, and his playby was a cartoon gif of a Skeleton dancing. Not to mention all the other people who use that skeleton skin, but don't have playbys. Are they in the wrong for playing skeletons too? Or is it just because of Undertale? Your point is kinda moot, there.
Mechinations are a bit different, and Dev has stated that they cannot be pregnant, but there's not really anything saying that they can't live a normal life or go insane. Hell, I thought the point of Mechinations (for the most part) was to be passing them off as humans for a long time, and that is indeed what most people do. Real-life playbys I don't think are against the rules, but I could be wrong. It's more like real-life face icons and playbys are just not done because almost the entire community uses anime playbys. And we know you're talking about Sora's demon baby icon, let's be real here. It's quite clearly an OOC joke. Michio literally had memes as face icons, so it's not that big of a deal. It's only a slight inconvenience to player who try to play for immersion.

I don't think this was honestly a situation that needed a big deal about it.