I use many browsers and don’t know of any service or addon that permits me to keep bookmarks in sync between one browser and another. There are online bookmark managers and I have an account at Pinboard.in, but that does not really solve the problem for day to day use.
But there are some points of light. It used to be that browsers were less standard in the way they handled bookmarks. I remember being able to import bookmarks into Opera, but not export them. There were even different formats for saving bookmarks. Perhaps there still are – I haven’t used Microsoft’s browser for many years. There were browsers that had folders in the bookmarks bar and those that didn’t.
Fortunately all the browsers that I use now permit the import and export of bookmarks as an html file, and all have bookmark bars with folders. So it’s easy to create a standard usage for bookmarks, back them up frequently and then import the backup file every time I start to use a new browser. To keep things tidy, I first delete any previous bookmarks, so I always have only the most up to date version of my bookmarks stash.
Because it’s so easy to maintain bookmarks now, it also makes sense to invest a little time in organizing them. So all my bookmarks are under folders and subfolders that I keep in the bookmarks bar itself. I have master folders for News, Email, Services, Forums, Social Networks, etc. and then subfolders of those. I don’t claim to have perfected the perfect organization yet, and of course it depends on my personal use case, but I can say that I’m a lot more organized than before, and it’s thanks to the fact that browsers themselves are more standardized in the way they handle bookmarks.
I still use Pinboard.in, but mainly for individual news articles that interest me, and to which I may like to refer later for one reason or another.
I don’t so much use sync between browsers on different devices. Mainly because I don’t really need that and partly because it means creating a cloud copy of everything and then trusting the browser company to safeguard that information.