I managed to get this working on Keycloak 26.1.4 by adding a mapper to the organization:* client scope.

I created a client scope called organization:*

Client Scope Fields

I then configured a new mapper for organization:* and chose Organization Membership for the mapping:

Client Scope Mapping Fields

I then added organization client scope as an optional type to my client, and organization:* as a default type.

Once I did this, the Organization claim appears in my access token, and multiple organizations display if my users belongs to multiple organizations.