You can use an Access table, but it is recommended you only use it through an OLE DB connection in order to maintain the data integrity. Don't consider a Personal Geodatabase and a non-spatial MDB file to be exactly the same. ArcGIS manages the data in a personal geodatabase, but you don't want it to manage the data in your tabular file.
You can use tables from a Microsoft Access database in ArcMap through
an OLE DB connection. OLE DB is a standard for sharing data between
applications, enabling you to view the Access database in ArcMap. To
maintain data integrity, you should only modify an Access database in
To connect to a non-spatial Access MDB or ACCDB file from ArcGIS you need to connect to is via OLE DB connection (as mentioned in your warning).
To do this you will need to add the Add OLE DB Connection to a toolbar in ArcCatalog. This button is found in Customize > Customize Mode > Commands, then a search for "OLE" should find it.
Click on your new Add OLE DB Connection button, and in the new Data Link Properties window make sure you're on the Provider tab, select "Microsoft Jet 4.0 OLE DB Provider" and click Next.
On the Connection tab, browse to select your database, and click Test Connection to check it connects properly. Once it connects click OK and name your new connection in Arc Catalog.
Now you can add your tables from an Access Database using Add Data and browsing your new OLE connection.
Access tables, like other tables without associated features, are only
viewable when the ArcMap Table of Contents window is listed by source.
See Connecting to a Microsoft Access database in ArcGIS and Understanding how to use Microsoft Access files in ArcGIS for more information.