What makes vector art have that mobile look? (And how can I avoid it?)

Flat shading and simple gradients mostly. Pixel art has complex brushes and 3D art has all the interesting light interactions making the gradients there more complex than radial or linear.

This may not be the entirety of the reason behind the noticeablle mobile "aesthetic", but vector-based sprites in those games looks like they're animated in real-time, with separate components (head, hair, arms, legs, clothing, etc) being stretched, rotated, and scaled as needed. See Fire Emblem Heroes, Fate Grand Order, Kirara Fantasia, and A Girl Adrift, among others.

I'm not sure of the middleware being used for most mobile games' sprite data, but as far as Fire Emblem Heroes is concerned, Nintendo/Intelligent Systems makes use of Sprite Studio (.ssbp file format), which basically defines the skeletons and animations for its character sprites. If you look at the image rips for FEH as well, the spritesheets for each unit are just paper-doll like bits and pieces.

If you want to avoid it, I can't really say anything objectively, but assuming that my observations can be applied to most mobile game aesthetics, it might not be that hard to avoid.

Which of these would you think have the "mobile look?"

Loot Rascals

Snakebird

Plants vs. Zombies

Neko Atsume

Grab a bunch of pictures for what else you would classify as one or the other and see what's the common thread.

I'd say it's very much an art style question, but there are only so many styles that work on 4 inch screens while not looking shit on a 13 inch screen. Basically, you have to create geometric volumes so you have colors and outlines even at very small scales.

There's flat shading and shiny gradients, but also really obvious bone/cut-out animation, thick line designs with polished, clean curves, and as well as that, lineless art can all betray that appearance.