Wrongfully held at Italian passport control for over an hour (EU citizen): can I get financial compensation?

I'd say the chance to get any kind of financial compensation is low. There is no compensation provision in Schengen Borders Code, and I doubt there is one in the Italian national law.

You can pursue the court case against them, but note that even Article 8 gives the border officials leeway:

However, on a non-systematic basis, when carrying out minimum checks on persons enjoying the right of free movement under Union law, border guards may consult national and European databases in order to ensure that such persons do not represent a genuine, present and sufficiently serious threat to the internal security, public policy, international relations of the Member States or a threat to the public health.

Note that there is no limitation on how long this "consulting" should take, and - unlike the rest of #2 in Article 8 - it is not required to be "minimal" or "rapid".

And of course they can claim they had suspicion about the validity of your documents - and until the validity is established, they cannot admit you as EU citizen. Here a point could be made that the border guard lacked proper training. But again its unlikely there is a law which would compensate you for having to deal with untrained government servant.

I think you are out of luck because in my understanding, the officer was right and you were wrong. Well, the officer is always right to begin with, but in this case, he really is.

You correctly state that as an EU citizen you have the right to move freely within the territory of the Member States and need nothing but an ID card. Ukraine is not a member state, so... you did not simply enter a member state's territory, but you did enter the Union's territory from God-knows-where (Ukraine in that case, and Moldavia before that, which also is non-EU) with merely an ID card. Which not only makes you "unusual" compared to most everyone else, but the ID card is also an item that is almost routinely refurbished for illegitimate purposes because of its sucks-so-much security features.

While it is indisputable that as a proven EU citizen you are finally entitled to enter and stay without limit^[1] -- what else, you have nowhere else to go -- it is perfectly legitimate to hold you back and do some plausibility checks when there's reasonable doubt. Because, well, you need to prove without doubt that you are indeed a citizen to have the rights that you claim.

I think it's highly unlikely (read as: impossible) that you will be able to prove that there was no reasonable doubt, so trying to get at the officer is pretty much a waste of your time.

I don't think you have any chance here.

Generalizing your problem:

The basic premise, opposite of what happens now would be that, if you were delayed in a security queue by the officer you would get a compensation for that.

The practical result of such rule would be that the border police would be pressured to avoid any delays to cut down any costs. And, in last resort the officer himself could be considered responsible for the delay, if he didn't have any good explanation. Suspecting someone would not be enough, because that would be applied in all delay situations and you would have a similar system. No government or organization will want that kind of pressure, because the result, can be a very poor security check. No one wants that responsibility in their hands.