I guess some of the confusion might stem from the use of mathematical lingo that you are possibly misinterpreting. In particular, the term "natural" means something like "obvious in a given context", and "no natural identification" doesn't mean that there is no possible identification.
You have to consider what he's saying here within the context of the previously discussed absolute Aristotelian spacetime. There was this notion that there is an absolute reference frame that you can refer to in order to say that a particular point at one time is the "same point" at some other time. That is, the assumption was that there's a way for everyone to agree on this - that's the "natural" identification (e.g. if you're given two arbitrary coordinate grids in two different times, if you can relate them to the absolute frame, you can unambiguously identify the corresponding points).
But, while you were reading the previous paragraph, have you stayed in the same place? You might say yes, but I might say that you haven't, since the Earth moved some 450 km on its orbit during that time. You and I named two different points in space as "the same spot".
Penrose is not saying that there is no way to connect points at different times, he's saying there's no single unambiguous "natural" way to identify points from these different $\mathbb E^3$s that everyone will agree on, which was taken for granted in the Aristotelian conception.