It wasn't that he was being strict, they just weren't his toys to play with. He was trying to be respectful of his friends things and though he trusted Enigma, if something did go wrong and there was an incident then he would have himself only to hold accountable and not her.
If she had told him, he would've reminded her of the consequences that would follow should she attempt to gain access to any of the suits or cars without Tony's consent. Both of them would be finding a new place to live.
He lead her to the garage and Jarvis allowed them to pass through since Bruce had access there, pausing once inside. "You can't drive but that doesn't mean you can't pick the one we take." He hated telling her no and was willing to compromise.