It was once very common for small independent multibrand boutiques in Italy to produce their own lines of clothing. Or perhaps more common still, to get third parties to make clothes for them and then sell them under their own name. But sometimes in-house production took place too. Some major Italian brands we know today started life in this way.
I often come across proto "collabs" between high-end boutiques and (at the the time) well-known brands. E.g. "made by XX well-known brand exclusively for XX high-end boutique, Roma" (or wherever). I've even found international collaborations of this kind, such as a Florentine boutique getting a British tie-maker to produce neckties in the UK (today these roles would more likely be reversed). The "Adrian's Original" label you found is probably an example of this. Although I admit I've never heard of either Adrian's nor this boutique before.
Generally I find very little info about these small-town boutiques from googling. Occasionally you'll come across a local newspaper article from the 90s mentioning that such and such a store or manufacturer has shut its doors, or an obituary for the owner. But rarely much else (although there are exceptions). I briefly tried looking in Italian and basically found the same business/phone directory listings that come up in English and nothing more. As Midge said above, in all likelihood the store closed too long ago to have left much trace in the digital realm
However, the Cavalli connection is interesting. Whatever you might think of Cavalli (me personally? trash), its an internationally-known brand with multiple stores across the world. It strikes me as highly unusual for a label of this caliber to retain the original name of a retail space they've taken over. Can you imagine Gucci, as part of their branding, keeping the name of the mom and pop store that previously occupied their new retail space?
Since at least the end of WWII (and likely even under Mussolini before that), Capri has been a playground for the extremely wealthy. Any stores that set up shop there will cater to their tastes (and budget). So we can already assume that Capri Sport was a classy establishment. Perhaps the kind of place that Kennedy Onassis might have popped into. The fact that Cavalli deemed it a sound business move to retain the Caprisport name suggests that the boutique was an extremely well-known Capri establishment, one that (within those circles at least) carries more weight than the internationally-recognized Cavalli label itself.
Unfortunately though, these are not the circles I move in, so I can shine no more light than this. What's clear though is that your skirt was likely to have been a high-end piece.
