"I have a white bunny."
Translation:Minulla on valkoinen pupu.
There's a little bit more to it; you're right, "on" absolutely does mean "is". But Finnish doesn't have a dedicated verb for "have" like in English. The way that Finnish speakers convey "having" something is to use a different construct, adding -lla or -llä to the subject followed by "on". -lla/-llä is a case ending which roughly means "on top of" or "at".
So "I have" is "minulla on" which literally translated would be "on me is". So directly translating the Finnish "Minulla on valkoinen pupu" word-for-word into English would be "On me is white bunny"