The income you and I have today is inextricably tied to the system of government, taxation, and trade that we are in. You can't talk about holding one fixed and changing the other in any meaningful way.
Now the toy examples - two guys on an island have some of their own problems. If it's just you and I on an island we won't bother with currency or payment (this is part of why it's a toy model). Their tends to be furious hand-waving between this sort of example and a "real" economy, and part of the reason for that is much of the mess as it scales up in practice (e.g. in actual history) tends to be creation of exactly the sort of institutions OP is trying to elide. I'm not sure I buy it, but there is at least an argument to be made that they are necessarily tied together.
You can talk about purely theoretical systems with a different basis - but they are completely different systems. Ones that nobody has actually worked out how to implement, for that matter, though there is lots of theorizing. Always worth remembering that the majority of economic theorizing seems to have a fairly loose connection to the observable world.
Fwiw: the desert island scenario, along with Smith's bartering villages are flawed enough to be probably best avoided.