I had the same issue and finally found the solution on Reddit…
When you’re not in your car, ask Siri to text someone (contacts name). Siri will bring up a list of one or more alternative apps to use along with iMessage. Mine was Google Voice (could be WhatsApp, etc).
I went into Settings>Siri & Search>scrolled down to Voice (or whatever app Siri has you choose between) Toggle the “Use with Ask Siri” to ON. Completely close out of the Settings App. Go back into the same screen and toggle the “Use with Ask Siri” back to OFF. Completely close out of the settings app again.
Give it a try. Worked perfectly for me. Didn’t make much sense since it was already toggled to OFF but once I turned it ON and the went back in and switched it OFF it fixed it. Hope it works for you as well.