If I change my Microsoft Account password, the Windows 10 Mail app still continues to send/receive e-mails and does not prompt me to enter the changed password.
That's normal behaviour and can't be changed. Your email password is linked to your Windows password, so when you logon, Mail is authenticated using that password.
I see from a previous post that you're worried if your computer was stolen they'd have access to your emails. They wouldn't however know your Windows password so they couldn't access Mail. If you have a portable device and you're worried about it being stolen while it's switched on then you could enable sleep to come on after a while of inactivity. Resuming from sleep also requires a password.