How to add podcasts manually to the new Podcasts.app on macOS Catalina
From what I found there is no way to add an audio file into Podcast application: as of macOS Catalina, it supports only subscribing to RSS feeds. The only way built-in way to listen to the separate podcast episode is to find it via application search and add it separately.

However, I found out that you can listen to any audio files including MP3 the same way you did in Podcasts in Books application by adding them there via File -> Add to Library... and then syncing them to iPhone.