No, it is not an extradimensional space
Instant fortress does not say it is or contains an extradimensional space. It says:
You can use an action to place this 1-inch metal cube on the ground and speak its command word. The cube rapidly grows into a fortress that remains until you use an action to speak the command word that dismisses it, which works only if the fortress is empty.
The cube rapidly grows into a fortress. That is, the cube already is in this dimension, it just grows. Compare this to other objects like bag of holding, handy haversack, or portable hole: they say that they are larger on the inside than from the outside, and they also reference extradimensional spaces, here for example from handy haversack:
This backpack has a central pouch and two side pouches, each of which is an extradimensional space. [...] Placing the haversack inside an extradimensional space created by a bag of holding, portable hole, or similar item instantly destroys both items and opens a gate to the Astral Plane.
If you homebrew a new item, that combines the features of one of those with instant fortress, for example by allowing items to remain within the fortress when you shrink it by putting those items into an extradimensional space, how that item works is up to you and the DM you work with. One way could be to use similar language (e.g. by putting the "instant fortress of holding" into another extradimensional space, or a bag of holding into it when it is shrunk, they both get destroyed and open a gate to the astral plane.)