Thanks for the input. What confused me was this blog describing the need to download some voucher from Migraciones to be eligible for the VAT exemption at hotels:

https://loyaltylobby.com/2024/06/26/...n-requirement/

In practice, I didn't have to do this last week. Paying with my foreign credit card was enough. But the blog post is why I felt the need to enter Argentina with a foreign passport instead of my DNI.

As others already confirmed, there was no special shared immigration processing. Uruguay exit at MVD, Argentina entry at AEP.