You may very well have spotted a familiar button that vanished from YouTube Music temporarily, following its debut on Android early last year. The service has supposedly reintroduced the repeat button while casting, according to 9to5Google.
In February of 2021, some users were able to use the repeat and shuffle features while casting YouTube Music to supported platforms, including the top smart displays. It became readily accessible a few weeks later.
Both capabilities, however, were quickly deactivated. The shuffle button was later added, however when you first started casting, the repeat button was still absent.
This works on the Android and iOS versions of YouTube Music, but not on the iPad app. The repeat button does not appear on the old Now Playing UI, which has yet to be changed to the Android tablet equivalent, for the latter class of device. In August of 2021, the iPhone version of the redesigned player was released.
The manual fix entails adding a tune to your queue numerous times or creating a temporary playlist, so perhaps this rollout will be permanent.