I'm not in New Zealand so I can't comment on the gun confiscations, but I can comment on why Commonwealth governments are interested if someone supports Trump or not since it is something that comes up quite frequently here; especially since the Brexit party took off.
For an American there's a lot of reason to care who becomes President of the United States. You come out to your polling booths and choose yourself, and this individual will spend the next few years enacting policies which could improve, damage or potentially even destroy your life and the lives of those around you. While the choice of President does affect all of us around the world, this influence is far less marked.
Imagine in America....Actually why imagine? Have you met
@Jacob Harrison, our resident monarchist/ex-Catholic theocrat? Jacob is very invested in who is the monarch of England, which might be somewhat relevant to one of the British subjects but to an American who rejects the rule of nobles? I'm sure you'd agree that's rather eccentric. Jacob likewise also is concocting a plan to overthrow Elizabeth II, abduct the grandchild of the heir of Liechtenstein and destroy the State of Israel among a range of other activities and political policies most of us would consider outside the norm and rather extreme.
Outside the US while not always the case an intense interest and appreciation of Trump rightly or wrongly frequently goes hand in hand with belief in conspiracy theories like Qanon which teaches and spreads some very extreme dogmas and slander that has led to violence. Again while also unfair it often tends to be a factor often shared with white supremacists and groups deemed dangerous like the National Front.
While it might be normal for an American to be fully invested and active in American politics, when done by non-Americans who really have no connection or horse in that race it's often a sign of an exceptional personality who might need watching. It might not be Trump's fault, but mass shooters both in America and outside of the United States have listed Trump as one of their inspirations and something they align with.
TL;DR: There's not really a good reason for a non-American to be that interested in the American president and it's usually a sign of something bad to be wary of.