Third-party browsers can no longer access Chrome's Sync API as Google mentions that they had to lock it up due to a security vulnerability. What this means if you're a user of a third-party browser on Android that made use of this API to sync all of your Chrome details and information, like bookmarks, saved passwords and the like, is that the browser's Chrome Sync option will no longer work. This doesn't mean that the browsers themselves won't work, but you won't be able to sign in to your Google account to get all of your Chrome-based options and settings, and if you rely on those, then a new browser app is likely what will need to be considered.
Google states that the Chrome Sync API was never officially supported by third-party browsers and that they have no intention of allowing third-party browsers to add themselves to a whitelist, which would essentially make the Chrome Sync option available again. Whether or not Google ends up coming up with a solution to allow the sign in capability with third-party browsers is unclear for the moment, so for now any users who want to be able to access all of their Chrome saved settings will simply have to use Chrome.
While most of the details are technical and the average user isn't likely to know what a refresh token is or what it's for, the main thing to make note of is that syncing your Chrome settings through a Google account sign-in is no longer available unless you're using Chrome itself. This isn't all bad news as Google has been making steady updates to the Chrome browser app on Android to where it runs faster and smoother than it has in the past, but if you simply just enjoyed the overall experience and features of another browser over Chrome then Chrome's improvements likely provide little peace of mind. If you still aren't up for using Chrome on Android, it might be time to consider exporting anything you can to another browser so you have all of the information you need or want, although that particular process is not necessarily going to be a quick one. It's also worth noting that this only affects third-party browsers based on Chromium that are for Android, and not those on the desktop.