... a thing or concept is referred to by the name of something closely associated with that thing or concept
The only example I can think of is tea. Tea can refer to the drink or the evening meal. I think the latter meaning only came about because an evening tea was accompanied by an evening meal and so the term itself became a shortcut for people to say "tea": I'm going to have tea.
It's safe to say that the drink (tea) has become a metonymy for the evening meal (tea).
I'm looking for a term to describe this process, when the original meaning of a word changes into something else, but both forms are used interchangeably?
I think semantic change or diachronic change could fit this definition but I'm not sure. Examples of semantic change from online give examples of words where the original meaning is no longer used and is replaced with its newer form. E.g. in Middle English meat used to refer to all types of food, where as today, meat refers to the animal flesh and food being reserved for all of the categories. Or gay meaning happy but now it's used in its derogatory form.