Wait, so your use case here is that you login to online banking and while you are paying your bills or whatever your computer dies, you pull the drive from the computer that just died and put it into the new computer, boot it back up all within 10 minutes, and then expect to still be logged-in? That seems exceptionally unusual, and logging into one account seems a small inconvenience compared to replacing your entire computer. tbh I'd be amazed if it even works now. Does Linux restore the complete memory state of a dead computer when you install the drive in a new machine?
Bank logins use session cookies that are cleared when the browser closes. Unless RAM is preserved you'll need to re-open your browser, so they'll be lost.
