Google has updated Assistant silently to add an option to select the default podcast provider. Setting a default podcast provider will set it as the player of choice across Google Home, Chromebook, Smart displays, Android 6.0+ TVs, phones, and tablets.
As of now, the settings menu in Google Assistant is only showing the option to set Google Podcasts as the default service provider, though other providers should show up sooner than later. Google does offer similar options for changing the default provider in Photos for Assistant but it does not support any other third-party alternative. Similarly, the Live TV section in Assistant only shows Sling TV as the service provider.
If you set Google Podcasts as the default podcast provider, you can then simply tell Google Assistant to play a podcast without having to specify the provider. If you do so without setting a default provider, Google will nudge you to do so. Google also highlights that its Podcasts app provides one with access to over one million free podcasts.
If you do not use Google Podcasts for listening to podcasts, you should select the 'No default provider' option, and then whenever you want to use Google Assistant for listening to podcasts, just add your podcast provider name at the end.