I agree there can be significant overlap. I do not agree that every case of resource allocation is a power distribution. As a case in point, there is a question that HN often asks -- what should I set my consulting rate at? This is inherently an economics question, and there is not often a power distribution discussion of whether the consultant has unique skills that let them charge what may seem to be outrageous sums, or if the consultant is freelancing in common skillsets.
Another point to be clear about is that economics is not solely concerned with resource allocation. Welfare consequences of policy, counterfactual reasoning, causal inference, algorithmic game theory and mechanism design, etc. are all research areas that are applicable to resource allocation but not only used for such.
Of course your consulting rate is a measure of power! The question about your rate is a question of how easy you are to replace. If you have skills (which are a form of capital, which is a form of power) or relationships that are hard to find then you have leverage to negotiate a higher rate. Power is at the heart of that question.
Whenever you are negotiating a rate or salary, you are forming a relationship, and the terms of that relationship depend entirely on your relative levels of power and leverage.
I guess I'll give you one thing though: whereas what two people want from each other can be complicated, power is not a scalar value like electric charge. But in many cases it acts like a scalar, especially if it takes the form of money, which is abstract and transferable.
Another point to be clear about is that economics is not solely concerned with resource allocation. Welfare consequences of policy, counterfactual reasoning, causal inference, algorithmic game theory and mechanism design, etc. are all research areas that are applicable to resource allocation but not only used for such.