One could also argue that dataframes are basically in-memory database tables. And in that case, S and SQL probably tie in terms of the creation timeline.
I think this is overblowing the similarities to matrices. Matrices have elements all of the same type, while data.frames mix numbers, characters, factors, etc. You certainly cannot transpose a data.frame and still have a data.frame that makes sense. Multiplying rows would not make sense either, since within one row you will have different types of data. Unless you have a data.frame that is all numeric, but in that case one should probably be using a matrix in the first place.
They still have their advantages with row/column labels, NaN handling etc. These are not operations I am speculating about by the way. I am most familiar with pandas and the dataframe there has transpose, dot product operations and almost all column operations have their correspondence in rows (i.e. you either sum(axis=0) or sum(axis=1)).
Statisticians are funny!