Okay so, i'll make a short explanation for people who don't understand Aizen's analogy and higher dimensionality.
In the context of aizen's transcendence / higher dimensionality
He decided to give an anaology to explain how his own existence works, so no it's not just an anology it literally applies to his own being in the context of how it's used. He states that just like a 2-Dimensional being can't interact with a 3-D dimensional being, they can't interact with him.
And directly uses "planes" and saying that he's in a "higher plane"
For which a "plane" means this
In geometry, a plane is a flat surface that extends into infinity. It is also known as a two-dimensional surface. A plane has zero thickness, zero curvature, infinite width, and infinite length. It is actually difficult to imagine a plane in real life; all the flat surfaces of a cube or cuboid, flat surface of paper are all real examples of a geometric plane. We can see an example of a plane in which the position of any given point on the plane is determined using an ordered pair of numbers or coordinates. The coordinates show the correct location of the points on the plane.
Meaning that to be in a "higher plane" of existence you'd need to have more than two-dimensions, and in this case 3 dimensions. which would already make him a higher dimensional being.
This absolutely makes sense, considering the previous analogy he gave about his dimensionality, his own being and that he's a transcendent existence.
The analogy also supports everything he claims, and stuff like him being able to destroy the kototsu (a being of reason that controls space-time)