Removing password from External Hard Drive

How did you create the external drive? What file system is being used on the external drive?

Make sure "ownership" on the external drive is set to "ignore":

If you don’t have permission to use files on a Mac disk - Apple Support

You should never mix or combine a backup drive with a data drive as you are just asking for problems and are more likely to destroy your backup which won't be available when you really need it. Besides if the data you are storing on the other partition is important, then that data needs to be backed up as well, but backing it up to the same drive is NOT a backup since if the drive fails you lose all copies of that data.

Purchase two drives, one for extra data storage and one dedicated to backing up your computer and the other external data drive.

My only guess for why you are being asked for a password for an unencrypted volume is because of a permissions issue which should be resolved if you select "Ignore Ownership" of the volume of the data volume if I understand that setting correctly. Otherwise you need to make sure your user account has read & write & execute permissions on the folder(s) where the data is being stored, otherwise you will be prompted for an admin password to access those restricted area. The "Ignore Ownership" of an external volume I believe is meant to eliminate this problem.