Creating and distributing similar emoji, after you have seen the original images, is surely a derivative work under copyright law, which requires permission from the original copyright holder. The images have apparently been licensed under a permissive MIT license, which would give you permission to make a derivative work and redistribute it, but there are several things to be aware of:
As you identify, there is a LICENSE file but no formal guidance about which components the license applies to. However, the language in the README makes it pretty clear to me that this was intended to apply to the entire repo ("This is the whole frontend for Game Jolt... We wanted to make it open source...") and I hope a court would agree (but I'm not a lawyer).
Assuming the emoji are indeed covered by that MIT license, then you may use them, modified or unmodified, under the terms of that license grant. That requires that the "copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software." You may make a derivative but must keep the copyright notice and MIT terms attached to your images somehow. Keeping them alongside in a separate file, as is done in the original repo, is fine.
If you didn't do this when you shared your derivative, then you were in violation of the license's requirements and infringed the author's copyright.
The MIT license is permissive in that it allows inclusion of MIT-licensed work in differently-licensed work. It is quite common for MIT-licensed work to be included in someone else's proprietary closed-source work and in GPL/copyleft work. You may place your own copyrightable changes under another license, as long as you also include the MIT-required notices that allow use of the original work.
To put it another way, you may not relicense the original MIT work, but that MIT license does not forbid distribution alongside/containing differently-licensed modifications. You may rely on the MIT grant of rights to prepare a derivate (as long as you abide by MIT's requirements) and simultaneously offer your changes under CC BY-SA 4.0.
As for what can be done about GitHub, I'm not sure. Assuming you abided by the requirements of the license, you might be able to report abuse to GitHub (or, as an extreme option, possibly sue the author for false statements that have caused you harm). If you didn't include the required notices, then unfortunately you did commit copyright infringement, and the author took a surprisingly harsh but ultimately permissible response for your failure to do so. I don't know what GitHub's policy is about reinstating accounts after infringement.
If you cannot get your account reinstated, there are numerous other services that provide free, version-controlled hosting. Notably, Gitlab and Bitbucket are well-known direct competitors to GitHub that offer free service tiers. When you do move your work somewhere else, make sure you include the appropriate notices.
Finally, I don't know what transpired between you and this other author, but if this author had your account disabled as a first step, rather than first asking you to add the MIT-required notices, then they may not understand that they have given you permission to make derivates. If they complain even after you add the required notices, then you may need to explain to them the rights they have granted you by attaching the MIT license to their work.
You may be concerned about including the permissive language of the MIT license alongside your more restrictively-license CC BY-SA work, because you are afraid someone may use it under the MIT license only. This is a common point of confusion about derivates that use work under the MIT license: you must preserve the license and copyright notice, but you may offer modified work under more restrictive terms. In short, the MIT terms apply, but you you also add more restrictive terms on top of those terms that apply to your modifications only.
To clarify this, you might make a single LICENSING file that begins:
The files in this directory are the combined work of the Game Jolt authors and Hacks Norris, who has modified them. You are free to modify and redistribute these emoji yourself, provided you comply simultaneously with both licenses below:
- the MIT license, under which Game Jolt is offered, and also
- the Creative Commons Attribution-ShareAlike (CC BY-SA) 4.0, under which Hack Norris's modifications are offered.
If you cannot satisfy both licenses simultaneously, you may not redistribute them and/or your modifications.
Then include both sets of licenses and copyright notices, yours and Game Jolt's, in sequence within that file.