This curious explanation (se non è vero, è ben trovato) makes no sense for reasons indicated by you.
Of course, Ber (given name borrowed from German Christians) is derived from German "bear" and Dov was created by Jews centuries later, as a Hebrew calque of Ber.
The association of Ber to Issachar (the former became considered to be a kinnui for the latter that, in turn, started to be used as a shem ha-qodesh for Ber) appeared a posteori, when Ber already was a vernacular name used by Jewish men in medieval Rhineland.
We don't know the exact reasons for the inception of this association. I heard one theory from someone (most likely, Yitzhok Niborski): through the intermediary of the Middle High German verb bër(e)n meaning 'to bear' that in various Germanic languages (including German and English) has the same or close root sounds as the word for the animal 'bear'. The link between Issachar and the verb "to bear" would lay upon Genesis 49: "Issachar is a strong donkey, lying down between two burdens".
We can't exclude the possibility that this link is not native either but invented a posteriori (as the Spanish-based explanation you mentioned) and that the actual link appeared just because there was a need to replace "donkey", with pejorative associations in West European culture, with another strong animal: Ber, already used (because of the virility / force symbolism associated to this animal [as did Yiddish given names meaning 'stag', 'lion', or 'wolf']) was a good available substitute.
Best,
Alexander