In Kodi, the YouTube extension is already pre-installed. However, you need to configure it to be able to identify yourself and find all your account (favorite videos, subscriptions, etc.).
If you already have projects, click on the list of projects.
Click on New Project.
Enter the name of the project and click Create. You can give it any name you like.

API and Services > Library.
YouTube Data API v3 and click on it.
Enable.
API and Services > Identifiers.
Create Credentials > API Key.

OAuth Consent Screen.
External user type and click Create.
application name, user support email address and an email address at the bottom of the page and click Save and Continue.
The 2 email addresses can be different.
OAuth consent screen and create a test user by clicking Add users, then entering an email address.
PUBLISH APP.
Credentials then, on top of the page, click Create Credentials then select OAuth client ID this time.
TVs and Limited Input devices and click Create.
The given name of the app on the same page can be edited or left as is.

Client ID and the Client Secret code. Keep these aside too.
The following steps can differ if you start Kodi or YouTube for the first time.
Start Kodi.
Go to Extensions > YouTube.

Settings at the bottom.
API on the left.
In the middle, edit each line as follows:
API key: the API key you generated first.
API ID: the client ID you got second. Do not enter the .apps.googleusercontent.com part, it is not used.
API secret: the client secret code you got last.
Once you have entered all 3 pieces of information, click OK in the settings window.
A popup will ask if you want to run the configuration wizard. Select Yes.



Video suggestions. If a list of videos is visible, everything should work after your identification is done.
Log in. A popup will tell you to log in twice.



You will get a window telling you that the page is dangerous. Continue anyway to finish identifying yourself.

Lo más probable es que tengáis un problema de permisos. En las opciones de la extension YouTube, verificad que Inputstream.Adaptive está activado.