This doesn't solve your problem, but I can report that the desktop version works correctly. When importing additional documents, even from an institution I've already imported from, the program will alwasy ask for the userID and password needed to retrieve the documents.
Personally, I've had to do just that with Vanguard because one taxable account is under my own userID and the other account is registered under my wife's userID.
Just guessing, but the problem could be coming from an "autofill" function of the browser.