I think it's entirely reasonable to assume that the Doctor has absolutely no idea how old he is. He just guesses.
That seems plausible.
Given, that for us non-time travelling people, it is a simple matter of measuring contiguous elapsed time from our birth to the present.
For the Doctor, unless he keeps a timer, there is no measure of his exposure to the passage of time. He can't look at when he started to what the date is in his present, as that constantly changes.