A particular stream of payments with a particular risk profile is an asset that has different (present) value to different entities like any other.

I may not be using the proper terminology here, but would I be correct in restating this as:

The underlying value of the asset is just one component, but the vehicle the asset is traded through can change that value equation by adding or removing its own value.

No, I don't think that's correct. A security doesn't have its own value independently of the payment stream it represents, any more than the deed to a house has its own value independently of the house. But the same payment stream (or house) might be of different value to different people, so buying or selling it can create wealth.

So the risk and form that the payment stream takes (ie. bond coupons are different from rental property income) might both be $100 for example, but if someone is overindexed in real estate in their portfolio, there may be more value to them in the bond?

Yeah. A risk profile isn't just a percentage, it's also a question of what it's correlated with.