As far as I know, the homepod has, and had before the latest update, two ways of playing music. One is by receiving airplay streaming from any capable device, and the other is by streaming on its own your apple/icloud music libraries.
If the homepod streaming music on its own, you can control it’s player, like remote control on any hifi system, on the control centre music menu. This menu will show your iPhone player, and your homepods and apple TV players. You can select any of this device’s player as long as you’re connected to the same network.
Say that you ask siri on a homepod to play any song, and later you want to add to the queue another song, but you don’t want to ask siri to do it. You then open control centre on your ios device, select your homepod’s player (not your homepod on your iphone’s player airplay menu) and your music app will now control the homepod’s player and anything you play or add to the queue will be on your homepod’s own player, and not on your iDevice’s one.
If you airplay content from your iphone the homepod just plays what receives from it but is your iphone the device that manages all the streaming from internet and playing functions.
I’ve noticed that with the release of ios 13.2 on iPhone and 13.2.1 on homepod and the added handoff with homepod, if you’re playing something of apple music on your iphone and you handoff it to the homepod, this does NOT select homepod on the iphone’s airplay menu making the homepod the audio output for the iphone; what it actually does is to set the homepod’s own player to play the same thing on your iphone and starting it on the same spot, and set your iphone to control homepod’s player. (I hope you understand what i’m trying to explain). This happens (or should happen) only if what your iphone is playing is playable by the homepod itself (anything of apple music, your bought itunes songs and your icloud music library), if your iphone is playing spotify, then it will select homepod as the aiplay audio output for your iphone.
That’s pretty cool and well implemented, BUT now , if you select homepod from your iphone/ipad on ios 13.2 (not earlier) player’s airplay menu it will not set the homepod as your audio output, it will actually swith to control your homepod’s player. It will ONLY set it as the idevice’s output if it’s playing something not streamable by the homepod itself (say netflix, spotify, youtube, local songs on the music app...). If you are not playing anything it will just switch to control your homepod’s player, and then if you select on your phone’s music app something the homepod cannot play by itself (like local music thats not on your icloud music library, it will not play it). Only if your are already playing on your device something the homepod cannot play by itself (only apple music and icloud music library songs) and you then select the homepod on the airplay menu, it will actually set it as the audio output and it will play, being your device the sender of that audio.
Before ios 13.2 on idevices and 13.2.1 on homepod, if you selected homepod on your airplay menu, it will send all your device’s audio to the homepod via airplay; and you could also select to control your homepods own player. This now has changed like i’ve explained.
I hope you understanded what i’ve tried to explain, sorry for my english mistakes i’m from spain