I tested this in base three. For primes below 1000, primes were followed by primes ending in another digit 2.1 times as often as they were followed by primes ending in the same digit. But the ratio shrank as the number of primes grew. By the time I was up to primes below 1,000,000,000, the ratio was only 1.2.

That hints that it's an effect that asymtotically disappears...