As I understand it it doesn't really extend all the rights of the home to your vehicle, it allows you to defend yourself if someone is attempting to unlawfully remove you from your vehicle.(a) The individual against whom deadly force or force other than deadly force is used is in the process of breaking and entering a dwelling or business premises or committing home invasion or has broken and entered a dwelling or business premises or committed home invasion and is still present in the dwelling or business premises, or is unlawfully attempting to remove another individual from a dwelling, business premises, or occupied vehicle against his or her will.
As it stands right now private property owners still retain the right to choose what to allow on their property, even in your personal car which is on their property.
Bills have been introduced that would allow people to store legally owned firearms in their vehicles in the parking lots of businessess; SB 792 & HB 5302. Right now they're sitting dormant, if this is something you believe in you should contact your reps. and let them know your thoughts.