As far as I can understand, the fine is for having a prompt for 3rd party apps, but not apple's own apps. Then I'm not sure because even to me, the wording used by the authority is not entirely clear, but the issue would lie in a different treatment reserved for 3rd parties compared to 1st party apps

Yes, precisely, take a look at the summary document [1] at the bottom of the article.

> xii. As a matter of fact, revenues from App Store services increased, in terms of higher commissions collected from developers through the platform; likewise, Apple’s advertising division, which is not subject to the same stringent rules, ultimately benefited from increased revenues and higher volumes of intermediated ads

> xiii. Therefore, considering that Apple holds an absolute dominant position in the market for the supply to developers of platforms for the online distribution of apps to users of the iOS operating system, the Authority established that Apple’s conduct amounts to an exploitative abuse, in breach of Article 102 TFEU, that started in April 2021 and is still ongoing.

[1] https://en.agcm.it/dotcmsdoc/pressrelease/A561_SUMMARY.pdf

ATT isn't about a vendor tracking you across their apps (Facebook can still log you into all their apps at once). It's about using data collected by third-parties or sending data to third party trackers, which Apple doesn't do for their own ads.