This is a historic question for which I couldn't find an answer on google or on the math history books I have at hand.
The first fundamental form $I$ of a surface $S$ embedded in $\mathbb{R}^3$ is, indeed, a differential form: a choice of bilinear map $I_x\colon T_xS \times T_xS \to \mathbb{R}$ for the tangent space of $x \in S$ that is "differentiable" in $x$ in the suitable sense.
However, since metric considerations about surfaces goes way earlier than the language of differential forms (say, Gauss' Theorema Egregium dates from 1827, Cartan's paper on differential forms is from 1899), I assume either the first fundamental form didn't have a name, or was called differently, or "form" meant something else back then. In the 1902 Princeton translation of Gauss 1827 paper, what we now understand as the first fundamental form is referred to as the "line element", as Gauss manipulates the expression $\sqrt{Edp^2 + 2Fdpdq + Gdq^2}$ in the notation therein.
Who first named the first fundamental form?