I'm guessing if you only calculate based on the digits, the probability is going to be slightly different than the real one, because you only have a finite number of plates you can choose from.

I'm glad you enjoyed! It was a real game I played when driving around.

Is your real name also Seth? This is wholesome and hilarious

Yes, the other name in the problem is my sister's name :)

Sounds like the birthday paradox problem. Is it?

Nearly, but not 8 digits of precision worth.

Can you reuse a plate with 500?

The wording seems to strongly imply no; you need two separate plates with 500 on them.