Another confirmation that the problem persists with 13.6. I use a Sony DSX-A410BT car stereo and this started with 13.5.
The issue seems to be bringing things into the foreground, because I am able to make Spotify play music (I just tried it earlier, "Hey Siri play [artist] with Spotify" worked; I don't use Spotify much so I never tried just "opening" it) but getting directions with anything that's not Apple Maps just fails. Before 13.5, I often said something like "Hey Siri, directions to [wherever] using Waze" and it worked perfectly.
Bringing up Siri by pressing the side button makes no difference.
The only case where the car image would show up on 13.4 was when I used the car stereo's button to open Siri, rather than calling it up with voice.
I reported this to Apple's Feedback Assistant site as soon as 13.5 came out, but my ticket's been rotting there for over a month now, open and unread by anyone.
This *must* be a bug, because there's a setting for DND to be enabled "when connected to car stereo" and I have it on "manual". It makes no sense to only apply "DND While Driving" to Siri when the rest of the DND features are disabled. I still see incoming notifications, and preventing me from using voice commands certainly isn't safer.
Moreover, I may be connected to the Bluetooth but not be driving, so why should the phone decide what I can or can't do via Siri, without giving me a chance to override it?
I really hope Apple comes around with this, at least in iOS 14, because it's extremely frustrating *and* dangerous. It literally contradicts the whole idea of "DND while driving".