It seems unlikely, why would they bother?
You might imagine they would want to add things you might click on.

Are you logged in to Chrome when browsing?
I wonder if saving bookmarks now depends on having an account and being logged in.

If you have a recent version it actually shows your user name at the top right.
I didn't care for that at first but does mean you are aware which account you are using - e.g. other family members could have their own.