There are no superscript. It is called moving the carriage to make the “th” slightly higher. How old are you, 30?
That's not even necessary on this typeface. The number set has the tails 3, 4, 5, 7 and 9 extend downward from the base line a bit, similar to a "g" or a "y" to achieve a fancier look. The "th" is actually on the baseline.
You can see this on the document in the year "1961" where the "9" extends below the baseline and the "1"s and "6" in the year do not.
It looked like the top of the th aligned with the top of the 4, which would make the th a smaller font, like the th in “Rather”.
You are correct that moving the carriage was a technique in use with manual typewriters; but that is not what you are seeing here. This typewriter's numeral keys contains several numerals that extend below or above the baseline and topline, an antique typeform convention. The 3, the 4 and the 9 extend below baseline, and the 6 extends above the topline. Look at the 4 in the two other places it appears besides the "4th" of the birthdate and you will see that it normally extends below the baseline. The "th" is normal, not superscripted; it aligns with the baseline.