What is bilateral symmetry in "bilateria", an actual symmentry
You are missing that "bilateral symmetry" does not mean "the divisions are the same on some arbitrary plane" it means "the divisions are the same on one specific plane".

That specific plane for animals with bilateral symmetry is on the mid-saggittal plane that divides the body into left and right. You won't get symmetry if you bisect in any other plane. If you do get symmetry in a different plane, then you are no longer talking about bilateral symmetry but rather some other type of symmetry, such as radial symmetry, spherical symmetry, or biradial symmetry.

Note also that although many animals do not have perfect bilateral symmetry in their internal organs (humans, for one), they are still considered to be bilateral symmetrical, because the bilateral symmetry is the main organizing feature of the whole animal.