I believe the answer you've pointed to is incorrect and here's why. The Customer Community Plus license is an "enhanced" version of the Customer Community license. It's totally counter-intuitive that it wouldn't have the same features as a standard Customer Community license. See Salesforce Communities Licenses for a comparison.
That having been said, all one needs to do is "clone" their Customer Community Plus license and add additional permissions to it. Once you clone your license, you can add the permissions you're speaking of or any other kind of permission set that you'd like.
Additionally, I'm showing in my Dev Org that the default settings for Accounts in a Community Plus User profile is Read, Create and Edit for Accounts. You might want to restrict that if you haven't already. Clearly it surpasses the requirements for your custom object and allows the access you're speaking of.
That having been said, in your post, you've used the term "Customer Portal Plus" license. Is this a "Portal" or a "Community"? I'm not aware of a Customer Portal Plus License, only a Customer Community Plus License. Is this perhaps some kind of upgraded portal user license that your org has negotiated with Salesforce or have you inadvertently used an incorrect term?