For these idioms:
- "take a chance"
- "kick the bucket"
"of his own" could be add to first sentence, but not the second:
He took a chance of his own.
He kicked the bucket of his own.
I wonder why?
The second idiom is much more metaphorical, and the bucket is a single, abstract concept shared by everyone. There are not multiple buckets available to kick; there's only one, and because we're all mortal, we'll all kick it at some point. Because of this, it makes no sense to say of his own, because that phrase always contrasts with a similar object owned by someone else.
In the first, there are multiple chances available to take, and one always personally owns a chosen risk. It would be correct to say took a chance of his own even if that chance was compared directly against another chance of exactly the same type. For example:
Alice checked her parachute and jumped out of the airplane. Hearing her cry of excitement, Bob decided to take a chance of his own and followed suit.
Alice and Bob each own their own risk, even though they're both jumping out of the same plane at more or less the same time.