From the description I'd guess that the OP was trying to voice dial contacts that were stored in the car kit itself.
It seems odd if the car kit is unable to access its own features as a result of a phone update, so does the car kit have settings of its own (maybe controllable through the phone)? It sounds like a default action has been changed, presumably due to the way the updated phone presents itself to the car kit, which is why I wonder whether there's a setting in the car kit to define what pressing that button does?
I'm sure Google Assistant can do voice dialling. But personally I don't use Assistant so am sympathetic to someone who maybe doesn't want to, and also curious about a third-party accessory that previously had this functionality itself seeming to now force you to use Assistant. If you don't use either Assistant or a Google Search app or widget then disabling the Google app kills Assistant completely. I don't know whether that would be enough to make it revert to the previous behaviour, or whether it will still prompt you - on my phone it removed prompts to enable Assistant, but I don't use a car kit (don't use my phone in a car) so can't say whether it would change how the kit responds to this.