Unable to access Google Play because of account verification problem
Because this account is the primary account on the device, Google Play tends to use it as the default. You can switch which account to use by opening the side drawer:



You can see the other account icons that are there, and I have a dropdown next to my primary account. Clicking on that allows me to choose a different account.

But since you don't know the password for the primary account, I would suggest the following options:


Log in to the Google account on the computer and reset the password. There you can click on helpful links like "forgot password". Once you reset the password you can change the password on the device and then the account will be verified.
Remove the account completely from the device. If the account isn't used for email and such, which since the phone is not logged in to the account, I would guess this is the case. Go into Settings => Accounts and remove the account that you do not know the password for.