I should have gotten the new Wear OS Spotify app since at least one week already but it doesn't even show up in the watch Play store. Phone and watch run the latest version of Android and also the latest version of Spotify for my phone.
I have a Huawei Watch 1 and a Nokia 6.1 with the new Android 9 pie update
It works for me now, I downloaded the latest APK on my PC and installed it via wifi debugging. For me it's fine but I can understand that this would be inconvenient for most people and shouldn't be necessary.