The problem is that the situation of a self-employed worker in Italy and in most English-speaking countries is rather different. Even in the UK, which shares EU law with Italy, the threshold for registering for VAT is much higher. Many self-employed people in the UK who would be part of the "popolo della partita Iva" in Italy don't have a VAT number because they don't have a high enough turnover.

What they have in common with their Italian peers is that they all are engaged in unincorporated self-employment, often the first step before incorporation, i.e. setting up their own company in some form or other.

"The unincorporated self-employed" is of course a tad clunky in many contexts. If you don't have to be too precise, you could shorten it to "the self-employed".