This is a nice analogy but a bit misguided in my opinion. If we corelate this analogy to the subject at hand, I presume we have to assume that elasticity is analogous to curvature of space time. Therefore spacetime is the rubber toy being exposed to elasticity and spacetime curvature is the elasticity. I wouldn't attempt to argue wether elasticity or curvature is a "physical thing" By all means both of these phenomena I would concider "real" but we also concluded in many threads that "real" is such a subjective term that it doesnt make sense to use it in this context.
Now, as for the title of the thread (the question at hand) do we have a solid definition of "physical" ? We can safely assume that "mathematical model" is a well defined and simple concept - its a measuring tool. As for "physical", previous threads shown that its not a simple task to define it In fact I don't think anynody has, its more of a semantics game. If we define "physical" as something which has to have mass, energy, volume or charge than things like spacetime or the 4 forces cannot be considered physical. Yet in my opinion they are - semantics again. So in order to answer if spacetime is a physical thing and/or just a mathematical model we need to define "physical"
Although I've read and watched numerous media on this subject since I've come to this forum 6 months ago, I cannot find a clearcut answer. For me it remains an intuitive answer (which doesnt satisfy me btw) - "if it bends, its a physical thing"