Is identifying the genus of a fungus (mushroom specifically) enough?
Again, I would say: no, genus is not enough.

Another example: boletes (specifically, genus Boletus).

According to Wilderness College:


  One of the most common and well-known groups of edible wild mushrooms are the boletes or boletus species (Boletaceae) ...


But


  Many species in this group are edible, with only a handful being poisonous. The poisonous boletus species have red or deep orange pores. Outside of the few toxic species, some species of boletus are bitter or inedible. 


For example, see here for a list of edible boletus mushrooms (e.g., B. edulis), while some examples of Boletus species to avoid include: B. huronensis and B. satanas

An additional issue to classifying all species in a genus as "safe to eat," is that our classifications are not always reflective of real taxonomic associations. In other words, some groups are not monophyletic and their evolutionary (and therefore physiological/nutritional) characteristics should not be assumed to originate from the same common ancestor . 

This is apparently true of the boletes. From Wikipedia:


  Boletus has been found to be massively polyphyletic, with only a small percentage of the over 300 species that have been assigned to Boletus actually belonging there and necessitating the description and resurrection of many more genera 



See for example: Nuhn et al. (2013)1




1: Nuhn, M.E., Binder, M., Taylor, A.F., Halling, R.E. and Hibbett, D.S., 2013. Phylogenetic overview of the Boletineae. Fungal Biology, 117(7-8), pp.479-511.