iOS Browsers are technically apps. What's stopping iOS browsers like Chrome from showing the ATT prompt?

Technically, nothing is stopping browser apps like Chrome from implementing the ATT prompt. It's not clear yet what Apple's position is on this (if they have one), but it is a valid question because one could certainly make the case that Apple might feel Safari itself is liable for any 'information leakage' that might take place as a user is browsing the web. Apple's Intelligent Tracking Prevention (ITP) is likely a nod in that direction.