What is the reason that the son from the first wife does not inherit his father, while the sons from the second wife do inherit upon the father's death, and can this cause animosity between half-siblings?

What was read in the mentioned fatwa is incorrect. Allah Almighty has detailed the rules of inheritance in the Quran for children, both male and female. There is no text that specifies inheritance for some children to the exclusion of others, unless there is an impediment to inheritance. A man's children inherit from him all of them, whether they are from one wife or from multiple wives. This is a matter of scholarly consensus.