The doctorate in Italy (dottorato di ricerca) was officially instituted in 1982, and the first courses started in 1983, so it was definitely not a common thing in the 1980s. People even got full professor positions without having one (since it did not exist).
Post-secondary education was not standardized across nations then like it is today; Italian scientists did their first research experiences with other research contracts, 'assistantships' (assistente) and other early-career positions that just weren't formally a PhD and did not end with discussing a thesis.
Regarding that CV, the most likely explanation is that "post-doctoral fellowship" is a liberal translation of the name of the Italian position she held. Note, though, that another non-standard practice of Italian academia is that people can legally be called dottore as soon as they get any university degree, even a 3-year bachelor. (although in the 1980s university degrees came after a single 4- to 6-year course, which was roughly equivalent to a BSc+MSc course with today's standards). So maybe in the eyes of the translator that position was "post-doctoral" because she was already a dottore (although not a dottore di ricerca).
As for your last question, it depends on the exact kind of position, but for instance I think you can still apply for a RTD-A (non-tenure-track assistant professorship) without a PhD [EDIT: not anymore since 2017; thanks @MassimoOrtolano for the correction].