How a discrete $Z_2$ symmetry removes flavour changing neutral current from Two Higgs Doublet Model?

It is not true that any $mathbb{Z}_2$ symmetry eliminates the flavor-changing neutral currents (FCNCs). A more accurate statement is that some $mathbb{Z}_2$ symmetries eliminate the tree-level FCNCs. By assigning the quarks and leptons appropriate $mathbb{Z}_2$ charges, one can avoid that two Higgs doublets couple both to $u$- and $d$-type quarks. The following table, which I found in this link, surveys the possibilities:

Scheme (i) is like the standard model plus a Higgs that does not couple to the fermions at all, so it is fine. The dangerous cross terms are also avoided in schemes (ii) and (iii) but not (iv).

As for "How will it differentiate among different flavours?": the simple traditional models don't, they only distinguish the $u$- and $d$-type quarks and leptons in their $mathbb{Z}_2$ charges. Settings (i)-(iii) work fine, i.e. do not require additional ad hoc assumptions. On the other hand, models in which the couplings are flavor-dependent are usually less simple.