Yes sharpie markers are permanent when written on plastic.
By permanent I mean that the sharpie ink will remain on the plastic without simply washing off from water and soap or just water and moisture.
So you can write on plastic to leave a permanent marking on the plastic and it will not come off at least without use of other chemicals and cleaning products.
I have however removed sharpie from plastic by using some acetone based nail polish remover and it worked just fine.
So in that instance the sharpie was not really permanent because it could be removed.
But unlike when writing on clothing with sharpie where it will come out in the wash the sharpie takes effort to remove from plastic.