You'd be surprised. Tigris is the latinized version of the name of the river in ancient Greek (τίγρης) which also means tiger in ancient Greek.

The common underlying etymology is an even older into-european term translated roughly as "sharp" or "pointy" (in the case of the tiger I guess referring to the teeth).

From a biblical etymology page:

The name Tigris shares its root with the word "tiger" (more precise: the word "tiger" and the name Tigris are identical in Greek). That means that in deep antiquity the tiger and the Tigris had signature qualities that were comparable and from which both derived their name. The word tiger and the identical name Tigris both come from the Avestan word tighri, which means arrow, or the more general tigra, which means sharp or pointed.

In French, the river and the animal are both "tigre"