When did people start thinking of elliptic curves as groups?

The first mathematician who talked about groups of points on elliptic curves (in the sense of Galois, i.e., in the modern sense of the word group) was Juel [Ueber die Parameterbestimmung von Punkten auf Curven zweiter und dritter Ordnung. Eine geometrische Einleitung in die Theorie der logarithmischen und elliptischen Funktionen, Math. Ann. 47 (1896), 72-104]. Poincare, in his important article referred to by ThiKu, does not use the concept of groups (this is the point of Schappacher's article). Even Mordell proved his theorem, namely that the group of rational points on an elliptic curve is finitely generated, without using the notion of a group: this was only done by Weil.

For a long time, most people interested in elliptic curves regarded them as a variety of diophantine equations, and preferred thinking about secant and tangent methods instead of group operations. Only when it became clear in the 1960s that the rank of an elliptic curve could be computed by applying all kinds of homomorphisms and determining the orders of kernels and images of such homomorphisms, the group theoretic point of view became indispensible.

H. POINCARÉ

Sur les propriétés arithmétiques des courbes algébriques

Journal de mathématiques pures et appliquées 5e série, tome 7 (1901), p. 161-234.

http://sites.mathdoc.fr/JMPA/PDF/JMPA_1901_5_7_A7_0.pdf