You don't have a tag for the resident pass. It goes by your license plate. The actual tags that you hang are for visitors. So if you registered your plate you shouldn't have to hang the tag.

Residential parking passes don't require ct plates. The guide linked here has instructions for cars not registered in New Haven. https://www.newhavenct.gov/government/departments-divisions/transportation-traffic-parking/parking-enforcement/residential-parking

Resident passes are a sticker. Visitor passes are intended for if you're having visitors for a shorter time, like sometime is coming for the weekend and needs to park. They hang from the rear view mirror.

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Technically yes, you need to deal with the tax assessor, even short term with out of state plates. In the past, the city has been aggressive about catching people who live here but keep their cars registered elsewhere. I have no idea if they have the manpower to strictly enforce this rule now. If you park on a street that’s heavily enforced, I’d imagine they’d eventually become suspicious. Nosy neighbors can also tip them off (less likely to happen someplace with a lot of temporary residents like East Rock imho). But the city doesn’t have a huge staff and resources are tight, so hard to say. As for the visitor pass, a quick search isn’t coming up with rules about how often or for how long a visitor can use a pass. Indefinitely? Maybe! The details are unclear.

What type of visitors pass? I’m not criticizing you, it just seems vague.