What do we call the “rd” in “3ʳᵈ” and the “th” in “9ᵗʰ”?

It's an ordinal indicator:

In written languages, an ordinal indicator is a letter, or group of letters, following a numeral denoting that it is an ordinal number, rather than a cardinal number. Historically these letters were "elevated terminals", that is to say the last few letters of the full word denoting the ordinal form of the number displayed as a superscript. The exact letters used vary in different languages.

(source: Wikipedia)

In the case of 9^th, the superscript shown is indeed called the 'ordinal indicator' as Matt Gutting has noted in his answer.

The addition of -th/ -eth relates to numbers 4 to 20 (and similarly,) and is a suffix to the cardinal number.

However, as in the second and third examples, the rd & st simply come from the right-end of the word for the ordinal number:

3^rd: thi rd

301^st: (three-hundred-) fir st (shouldn't that be 301 th ?, I'm not going there).

Of course, in general, we call all these superscripts 'ordinal indicators,' and "suffixes," 'ordinal suffixes.' (We can see that there's no suffix as such until we come to 4, as we have ordinal names.)

First, these abbreviations are not onomatopoeia.

In fact, your question is about writing rather than sounds.

In the written number 2^nd, the letters ^nd are the superscript.

More generally, these letters form the ordinal indicator in english.

These are ordinal suffixes.

Ordinal formatter: add ordinal suffixes (-st, -nd, -rd, -th) to numbers. finzi.psych.upenn.edu