The magic of multi-library borrowing is Overdrive's Libby app, not Overdrive's website. It's not intuitively explained anywhere. I borrow from three libraries, and they all download onto my Kobo Aura One because they are all linked to my Libby app.
Download Libby >
https://meet.libbyapp.com/
Set up your account and link to Overdrive.
Add libraries in Libby.
On your Kobo, make sure you are signed into Overdrive using your Overdrive account, not a library account.
Borrow books through Libby.
Sync your Kobo.
Tada!
One of your libraries will be the main library. This library will be the only one that your Kobo can interact with as far as renewing, returning, and borrowing from the Kobo store. I get around this by using Libby to manage all my books across the three libraries.
Does that help?