Headlines and a link make for a shitty notable.
>the straw is called mate
Not here in Brasil it's not.
We call it a bomba and according to wiki link you posted, the word is bombilla in Spanish.
>the cup is called mate
Again, not in Brasil.
We call it a cuía.
I understand about 60% of Spanish based on my Portuguese, and 100% of the dialect spoken in Galicia, even though they spell words differently.