Not all domain names are created and distributed equally @meter. Many extensions that have become popular are actually or were initially assigned to a country. The specific TLDs assigned to a country belong to a subcategory called ccTLDs (country-code top-level domain names). I remember when the .TV marketing rights (originally assigned to Tuvalu) were sold. In particular the .AI domain is the country code domain extension (ccTLD) for Anguilla, a small island nation in the Caribbean Sea.
Because extensions like .AI, .CA and .US are assigned to a specific country or entity they each can be managed and governed differently. You will find that ccTLDs can have specific requirements on who can transfer, acquire, register and sale them.
Looking at the .AI ccTLD specifically it looks like Offshore Information Services controls the management of that domain extension. There doesn't seem to be any restrictions to who can register a .AI domain and they can be purchased worldwide. Only Offshore Information Services (or those they assign) can sale this specific domain extension but you can purchase it online. I hope that helps?
roy darling *my posts seem a lot shorter in my head