No, the rope trick space is it's own demiplane
"Demiplanes are extradimensional spaces that come into being by a variety of means and boast their own physical laws. Some are created by spells. Others exist naturally, as folds of reality pinched off from the rest of the multiverse. Theoretically, a plane shift spell can carry travelers to a demiplane, but the proper frequency required for the tuning fork would be extremely hard to acquire. The gate spell is more reliable, assuming the caster knows of the demiplane.
A demiplane can be as small as a single chamber or large enough to contain an entire realm. For example, a Mordenkainen’s magnificent mansion spell creates a demiplane consisting of a foyer with multiple adjoining rooms, while the land of Barovia (in the Ravenloft setting) exists entirely within a demiplane under the sway of its vampire lord, Strahd von Zarovich. When a demiplane is connected to the Material Plane or some other plane, entering it can be as simple as stepping through a portal or passing through a wall of mist."
Note that Mordenkainen’s magnificent mansion is given as an example of a demiplane created by a spell. This implies that other spells that act similarly also create demiplanes.
From MMM spell description:
You conjure an extradimensional dwelling in range that lasts for the Duration. You choose where its one entrance is located.
From rope trick description:
At the upper end of the rope, an Invisible entrance opens to an extradimensional space that lasts until the spell ends.
These spells are written in the same way and I would suggest they are treated the same way.
It is also worth noting that demiplanes are being defined in the passage from the DMG as "extradimensional spaces that come into being by a variety of means and boast their own physical laws. Some are created by spells. Others exist naturally, as folds of reality pinched off from the rest of the multiverse."
A rope trick extra-dimensional space meets this definition.
I believe the distinction is made so that if you wanted you could have an extra dimensional space that was on the same plane but it would not be separated from the plane. Perhaps you walk into an area within the plane where things behave differently but it is not "pinched off" from the material plane so things can pass through easily, without passing through any sort of entrance, portal, wall of mist etc.