Also in Italian. "Varietà" is the term for both.
Starting from this, I looked at the Italian Wikipedia webpage for varieties which, at the end, has this remark about the origin of the term:
In italiano si traduce con varietà il termine tedesco Mannigfaltigkeit, che compare per la prima volta nella tesi di dottorato del 1851 di Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Riemann si pone il problema di introdurre delle "grandezze molteplicemente estese", aventi cioè "più dimensioni", e le definisce usando quel termine.
Analizzando il termine come parola composta, Mannig-faltig-keit, si riconosce in essa un parallelo con il termine latino multi-plic-itas, sicché lo si potrebbe tradurre letteralmente come 'molteplicità'.
That can be approximatively translated as
In Italian we translate with "varietà" the German word "Mannigfaltigkeit", which appears for the first time in the 1851 doctoral thesis of of Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Riemann introduced some "dimensions with multiple extensions", having "more dimensions", and he defines them using that term.
Analiying the term as a compound word, "Mannig-faltig-keit", we can identify a parallelism with the Latin word "multi-plic-itas", so that it can be literally translated as "multiplicity".
So, at least in Italian, the term comes from "variety" and "multiplicity" in the sense of "diversity".