"On my own" is about independence: I live on my own; I can cook breakfast on my own; I found my way to the library on my own; I traveled to Italy on my own. "For my own" is possession: I want it for my own, which underlines the fact that it would not be shared. Often you hear it with "very:" I would like a puppy for my very own; if you have something for your very own, however, that further means you are proud of it or cherish it: I got a pair of brand-new ice skates for my very own.

...Backing up a little on that one. I've dug around a bit more and, to my surprise, I came across some examples of 'I want xxx for my own'. 'For my own' appears to be some kind of 'fossil' that only exists in that one phrase. I wouldn't recommend using it, though, as it's very low-frequency and seems to be non-standard.

"on my own" is used to say you're going to do something by yourself for example "i am going to do this on my own" or "i am going to shops on my own" i cant really think of any situation you would use "for my own"

I don't think there is such an expression as "for my own" in English.

Sample1 on one's own: Hi, boys, I hope you may accomplish the level test on your own (by self) without discussion with your peer Sample 2 for one's own: Most of the capitalists in the market just care the vest interest they hold, in other words, they make profit for their own. Wish that will help you