The explanation:

https://twitter.com/AricToler/status/1841560036201361799

So, Trump is using a branded notification for this -- thanks to @l18cp for pointing this out.

It's basically like a tweet (or semi-public DM, I guess) that won't appear in your timeline or searchable.

Mystery solved!

https://business.x.com/en/help/campaign-setup/creative-ad-specifications

Link to the specific section of the linked page