Does assigning DOI numbers signal anything about the authenticity of a journal? Why do some journals not assign them?
DOI, by itself, certainly does not add "authenticity". 

The Digital Object Identifier is, by design, meant to be a method of producing and providing a persistent actionable identifier to entities. To put it in plain English: the DOI is an effort to prevent link rot, and importantly, does so by not only providing a registry of the numbered items, but also providing a mechanism for accessing/retrieving those items. The Wikipedia Article on DOI also explains pretty well what it is for. Notice that DOIs can be had for many more than just journal articles. In science, a growing practice is to use DOI referring to datasets. 

As to why a publisher may decide not to participate in the DOI scheme: for starters, it does cost money. This may have contributed to the correlation that sham publishers typically do not make use of the DOI service.