Shouldn’t we use these instead of this in the above quote?"During our study we discovered two new species: Hydnum ovoideisporum and Hydnum vesterholtii, which belong to a genus commonly known as 'ox tongue'. Although many of this species' mushrooms are consumed in many parts of the world, paradoxically there have been few recent serious studies that attempt to clarify what species exist and the differences between them," as explained by Ibai Olariaga Ibarguren, lead researcher of the study conducted from the University of the Basque Country.
Two new species of mushroom found in the Iberian Peninsula, Spain
"Many of the mushrooms of this species are consumed in many parts of the world."
However, the species would have to have sub-species, varieties or cultivars, otherwise you couldn't have "many mushrooms of this species".
I believe they actually mean "Many [species] of this genus of mushroom ..." or, as you say, "Many of these species of mushrooms ..."
I assumed they were referring back to the two new species that have recently been discovered, mentioned in the very first sentence. If that is the case, then "Many of these species' mushrooms are eaten ..." would be grammatically correct as far as "this/these" is concerned. However, I don't see the need for it to be possessive. I would have said "Many mushrooms from these species are eaten ...", as long as they mean that a large number of mushrooms belonging to those two newly-discovered species are eaten ...
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.