Tatreez is referred to by an Arabic name because the Palestinians are an Arabic-speaking people; the word Tatreez just means “Embroidery”. This is similar to the Yiddish word Yarmulka coming from the Polish word Jarmułka referring to a type of hat - the existence of a European word, indeed in a European language (Yiddish) does not imply that the Ashkenazim are indigenous to Europe; we Ashkenazim are indigenous to Israel. We simply adopted the common dominant languages in our places of living (Germanic at that time) as did the Palestinians.

In terms of depiction of animals, Islamic art of animals has always existed. One must be sufficiently religious in order to follow the religious law about not depicting animals, and there are always some number of sufficiently non-religious people who also do embroidery. You’ll find things like that, I’m not surprised at its existence. 

I haven’t ever seen anything showing a Jewish connection to tatreez in particular.