To my surprise, your advice helped. But what's interesting is that I tried to change the connection types (AI Key and also Microsoft Entra ID) but it didn't work before until I deleted the connection(in AI Studio) - to AI search and added it again (private endpoint was active).

I also left only the API key type enabled in the AI ​​search (settings) - as the only one possible method (there was another recommendation in the Microsoft article)

Thanks for the recommendation!