What is a Scrum product and Scrum value

You forget that not all products are made for external customers.  Many products are created to be used internally so the value may not be measured in monetary terms.  My wife works for a non-profit health agency and they frequently use other methods to determine the value their products provide.  One such is how many units of blood that they processed helped to save a life.  (And yes, they do use Scrum-like processes in their labs. I know because I helped implement many of them.)

And, most importantly CUSTOMERS-which means "even more abstract" products are commodities.

In the example I gave from my wife's work, they do not have customers.  They have patients.  Yes, the products that they produce are commodities but not the type that are used to realize profit. 

You mentioned the definition of "customer" from the Merriam-Webster dictionary.  That is only one of the definitions.  The other is "an individual usually having some specified distinctive trait". I will admit that a patient is an individual usually having some specified distinctive trait.  So, yeah they are customers. 

The Scrum Guide definition of Product you provided is correct.  But you forget about the "service...or something more abstract".  I worked at a company where Scrum was implemented in the Human Resources division.  They provided "services ... or something more abstract" to the internal customers (i.e. employees).  That same company had a large organization that coordinated trade shows and many other events.  And yes, they used Scrum also.  I would say that a trade show, fundraising event could be considered "even more abstract". Not everything is tied to revenue in all organizations.  Sometimes value is more heart felt than wallet felt. 

Your example seems a matter of politics maybe; because you don't see hospital patients as a "customers". Possibly - because you are describing operations in UK where healthcare is charge free on patients side(funded by government).

In places with privatized healthcare (like USA and many EU countries) the patients are clearly " the customers" who are purchasing the healthcare services from the doctor or the hospital, so this brings whole operation to monetary scale.

In your example the monetary aspect is less visible because, as I have said, its not patients but the government who is footing the bill. But still, every medical service has it price, customer value and investment, and its profit and loss account; which in case of Britain is reflected at the balance sheet of the hospital or balance sheet of the whole NHL service

So in fact your wives efficient use of Scrum is either increasing revenue of the branch, or helps the organization reduce charges, which makes in a monetary operation?

By saying "monetary operation" I don't necessarily mean commercial companies who work for profit(which was a core of Silicon value where Scrum was born and began growing).

Scrum can be used by a charity or state operated service provider who is providing "free"(taxpayer funded) services to the people

But any charity and state run agency still has a balance sheet, gain and loss account, funding and and cost, and those are the aspects which generally reflect the success or failure of operations.

As for internal operations they are routine in many companies an big corporations, where one branch or department is often a supplier for another.  But internal exchange of products and services (or other more abstract commodities, who are beneficial and hence have market value) is always reflected in the balance like an internal trade, and is generally not much different by its idea to external sales.

But this brings us to the much wider question then Scrum, about the borders and limits of monetary aspects of life and exchange between humans

Some people argue that any exchange between human beings has a monetary value, some other argue that only actual sales for currency have.

I live in the United States.  And if you work in healthcare you see things differently.  While your vision is based upon commercial software development, not everyone sees it that way.  With all due respect, your vision is skewed towards the monetary aspect.  I see value in improving employee morale, in making work easier, in providing services that help people better their lives.  I'm sure that all of it can be tied to money in some way.  But the value that is derived from a product does not have to be monetized. Value can be felt and recognized in many ways by different people.  Some organizations do not choose to measure everything in currency. 

As I have said question if all human exchanges should or should not be measured by money goes beyond borders of Scrum. In fact whole modern day politics, including USA one is rooted in this discussion

But coming back to Scrum, if one is refusing to measure the product value in money, then some other metrics, and exact, not the abstract one, should be introduced....