The 10:03 Mystery

The reason for seeing the numbers as often as you do probably relates to you wondering what it means.

In general, I'm unable to use numerology to calculate personal meanings. But this is what I think it is:

The mind doesn't like mysteries and tries to get them solved whenever there might be a chance to do so. Therefore, whenever the time is 10:03, your mind tries to direct your conscious attention to the clock. From what you say in your question, it often succeeds.

Somewhere, somehow, the mystery got started. Perhaps when you saw your birth date on the clock the first time, you wondered what it meant. The first time that happened, it became a mystery, which the mind is compelled to solve.

Like attracts like. Therefore, you're somewhat more likely to see the numbers of your birth date on the clock than you are to see another specific set of numbers with no relation to your numerology profile.

But the tendency probably isn't strong enough to make you see the time as often as you do. Which is why I think the reason is the mystery of the whole thing.