On Discord, we can’t upload SVG custom emoji because vector drawings can potentially break stuff.
SVGs can be arbitrarily complex to render (imagine an SVG file containing instructions to render hundreds of thousands of bezier curves or something). Which means that someone could craft an SVG that would slow down/freeze the client while it tries to render - and it’d be pretty hard to develop a reliable heuristic for detecting this.
Builtin emojis are OK to be SVGs as they’re obviously not malicious, but we can’t guarantee this for custom emojis.
I suppose that would be the same for any app that allows users to upload SVG files, but I don’t know if some restrictions can be set to limit potential issues.