Does italic math originate from TeX?

No, it was a standard convention. See, e.g., “Uniformly Convex Spaces” by James A. Clarkson that was published in 1936.

I've never noticed any math books with an upright font for variables. I think italics have been very standard for a very long time, at least in American books. Here are three old American books that use italics:

• Hutton, A Course of Mathematics, 1807

• Davies, Elements of the Differential and Integral Calculus, 1836

• Granville, Elements of the Differential and Integral Calculus, 1904

Also:

• L'Hospital, Analyse des infiniment petits, pour l'intelligence des lignes courbes, 1716