Referred to by some as Hyperspace
The subdimension is a plot important place in Tokumei Sentai Go-Busters.
It is the place where the Transport Research Center and Messiah were sent 13 years before the start of the story. Hiromu's parents, Youko's mother, and Ryuuji's Sempai were in the Transport Research Center when this happened.
In the present day, it is the place from which the Vaglass transport their Megazords using the Metaroid as a Marker.
It is learned that Jin Masato survived the transport and is alive in the Subdimension. He sent his Buddyroid, Beet J Stag, to act as a Marker. This allows Jin to send an avatar to the real world, thus allowing him to fight as Beet Buster. J's Marker System also allows them to transport BC-04 and SJ-05 from the subdimension to this dimension.
Naming as Hyperspace
This comes from Nakamura's computer displaying that it's a Hyperspace Monitoring System. This makes sense since Nakamura was trained to monitor the subdimension. There are times like in episode 20 which show a clear picture of one dimension labeled as Hyperspace, connected by a wormhole to another plane that's labeled as Properspace.
Hyperspace in the context of Nakamura's equipment might just be referring to monitoring things that are in the middle of teleportation. This would fit with the usual definition of Hyperspace in sci-fi. It just really depends on if the creators of Go-Busters intend for the subdimension to be the space intersecting our dimension that's used for teleportation. If so, that would fit a commonly used sci-fi definition. Hyperspace.
Hyperspace in sci-fi would refer to something in the middle of teleporation or hyperspeeds. A good example of this would be the Millennium Falcon when it's going faster than light. In that sense, the analogy in Go-Busters would be Hiromu when he does his super speed move, that would be Hiromu when he's in the middle of hyperspace.
However, this does not appear to be how the term is used in the series. If this were the case, then the weapons and other items teleported using the Transpod would need to travel through hyperspace to arrive at its destination in our dimension. So far in the show, we have seen no evidence of items appearing in the subdimension then reappearing in the current dimension. Megazords also seem to take some time to transport from the subdimension to this dimensioin, but that might be because of how much mass they have.
In most likelihood, the term Hyperspace probably does refer to the subdimension based on the picture from Nakamura's monitor. Both akuukan and Hyperspace are broad enough sci-fi terms that they can be applied without necessarily being wrong. It just comes off as throwaway sci-fi jargon.
Since the Subdimension that Messiah is in does appear to be its own self-contained dimension, T-N will continue to use the term Subdimension. The term Hyperspace does not conform with the western conventional concept of the known term, and so far the term only appears in the on-screen text, which has a habit of being arbitrary or incorrect, so it's probably safe to ignore it for the time being. There is also the possibility that Hyperspace is being used correctly, and thus only refers to things when they are in the middle of teleportation.