Certainly not a Fish vase, I'm really sorry to say. You're right to think first off about how the different pieces are made though as that, and the general appearance, are the keys. Fish have very heavy, wide and quite often very thick 'wings' that are gained from dipping a body into the pot, then flattened down on the marver and with paddles. This two part casing typically covers the entire underlying coloured body. The full process is listed in my book an perhaps the most representative Fish design is on the cover.
On your example, hot trails have been applied up the sides of the piece from the clear base. The trails are much thinner, near cylindrical, and are exactly that - applied trails. David's right - it would indeed be classified as a Lollipop vase from its shape, but also has the trails associated with the Side Stripe design. Throughout all of Mdina's history, experimentation has been rife - I've learnt 'never to say never' about shapes and sizes, which were combined, modified and changed. I've seen these 'Side-Stripe-Lollipop' combos before, but they don't crop up as often as some other shapes. They generally tend to be later, so 1980s or so, but I'd say yours was mid-late 70s from the colours. They do seem to be popular amongst collectors. Actually, thinking about it, this is the nicest one I've seen!
Finally....not to confuse matters....the only relation the Lollipop shape has to a Fish is that a Lollipop shape is the core of the Fish before it is cased!
Hope this helps,