As far as I know it's only illegal for minors under the age of 18 to purchase or carry a sharpie permanent marker in a public place.
The reason is that some minors such as teenagers may use the sharpie to deface things.
So in some places it's illegal for a minor to even purchase the Sharpie or permanent marker and you must be 18 or older to purchase the Sharpie.
I saw a boy of about 13 trying to buy a Sharpie at Walmart here and the cashier said she couldn't sell it to him because he was under 18.
So if you carry a sharpie and are under 18 then you might get it confiscated and get arrested or get a fine.
But an adult would unlikely get arrested or get fined for having a sharpie in their pocket or in their vehicle.
I keep a sharpie marker in my truck and car and have been pulled over with it in plain sight of the police and they never said anything about it.