There are two APIs from YouTube
The External API is for Developers and people while the Internal API is made for the web apps and other official YouTube clients. So, Those clients use the API Key you provided. The API key you created by linking your account is for the external API.
Using the External API key on the Internal API is not good, bruh!! The InnerTube API does not need any account linking or anything but is not properly documented anywhere... so, use it wisely...