This is analogous to elementary school "grades", where the article can be used.
What grade are you in?
--I'm in (the) first grade.
We can present the item in the ordered series using the definite article, in which case we're identifying its ordinal position within the series and ignoring any individuating characteristics of the item; or we can present the item without the article, in which case we're presenting the item as an entity unto itself, having its own identity outside its ordinal position -- it is not merely one of a series of identical items.
In the case of competition, where there are three positions on the podium, or in horse racing, where there are three moneyed positions, win, place, and show, we are looking at instances of the second scenario described above, where the item-at-position is not merely an item in a series of identical items but a thing unto itself: the winner gets gold, second place gets silver, third place gets bronze, and the glory associated specifically with that position.
P.S. So, if you ask a six-year-old child "What grade are you in?" the child might reply:
I'm in first grade!
I'm in the first grade.
Of the two, the first answer, without the article, is more likely to be given with a tone of pride and accomplishment and the second as merely a statement of fact.
P.P.S. Put abstractly, with "in the third place", the article specifies "third" and thereby foregrounds the idea of ordinal position, its position in the series; whereas with "in third place", the idea of "place" is foregrounded with its ordinal position in the series backgrounded to the status of an attribute.