The steps I took to remedy the problem are outlined below.
The first thing I did was go into the user's account and see what licences were given to them.
Either Azure Active Directory or the Office 365 Admin Portal can be used to do this.
My account has a Power BI Free License and a Power BI Pro License, much like in the sample below.
The Power BI (free) license was assigned to the user account initially, according to my understanding. Then came the assignment of the Power BI Pro license.
To fix this, I had to delete the user account's Power BI (free) license.
Following licenses were allocated to the same user account once it was upgraded.
I then asked the user to log in again, and they were able to see the App and the data it contained.