I don't have any explanation about why immigration never asked (when you came back into Japan) if you were still studying; maybe this meant the university never informed them that you completed your studies. (But it is not that immigration will try to track down everyone who has finished studying but has not left the country.)
The internship programs were unpaid, you still had a valid student resident status (though no sponsor), and you did not give it up when you left in August 2016, maybe because you thought you "might" return for another study program? (Did you know already you were coming back for internship?)
So the only thing I clearly can say you didn't do "right" was that you did not inform immigration that you completed your studies (no longer with the original university), which both the sponsor (uni) and the status holder (you) are obligated to do within 14 days of the change. And maybe another thing is (you don't say if you did or not but probably) that you did not inform immigration of the new sponsor (the intern employer, if I may call it that), and just made it look like you were still part of the initial study program but temporarily placed with those companies. Did the company where you did intern ever asked about your status, and if they needed to do anything for you related to that, like supplying a paper saying that you are an intern there? Maybe not.
I really don't know what to make of it, but just to say that you did not blatantly do anything wrong, you thought you could do unpaid internship while on student status (which is not wrong, if done as part of a study program), you did have a valid student status, no immigration officer ever ever asked anything of it, uni (maybe) did not inform immigration, your intern "employer" posed no question, etc., etc. all those things by chance came together. And you (by now) already gave up that status. And since you did not earn anything financially off the internship programs, and I assume you did no part-time job during your internship? So you did nothing that someone for example on Temporary Visitor status cannot do (provided Italians are given six months with Temporary Visitor Status, because you stayed in total 6 months).
So...can anything bad happen?
If you ever get a chance to work for a Japanese company in the future, you (or your employer) applies for a work visa for you on the same passport, your passport number will appear in their record, immigration might ask you about your past stay in Japan, you tell them (with evidence if you can) that after your initial studies you did completely unpaid internships (to brush up your skills or whatsoever) for a while, in the belief that student residence status allowed it, what can they say.
The ultimate decision is of course up to the immigration authorities, so if you ask me can they ever reject your visa application, my answer would be, yes, there is always that possibility. But seeing how things fell into place so far, "what can they say?" is my take. (Japanese with a non-Japanese family member, who used to work in various work environment, so have seen different situations involving the immigration authorities...)
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