It's still a 1/365 chance, there's only 365 available days.

You’re right, but that’s not the explanation.

It’s 1/365 because the specific date of the father’s birth is irrelevant. We’re actually only calculating the odds that the son was born on the same day.

In other words, the odds of the father being born on any day of the year which is 1, followed by the son being born on the same day which is ~1/365.

If you specify a specific day ahead of time, then OP is correct. Odds of the first birth landing on that date would be 1/365 and the odds of the second birth landing on the same date would be 1/365 so together the odds would be 1/133,225.

There’s only 365 available days, but there are 133,225 ways to select two of them at random.