If a molecule, or any conformation of a molecule, contains a plane of symmetry, then the molecule is not chiral (achiral).
The all hydrocarbon catenane will be achiral. The molecule contains a plane of symmetry; picture a plane that contains one of the rings and bisects the other ring. In the following image, you can see that the plane of the screen contains the ring on the right and bisects the ring on the left.
You might say that if we replaced the circles with cyclohexane rings, because of their chair-like shape, the plane of symmetry is lost. That is true, but you can flatten one of the cyclohexane rings and the plane of symmetry is restored. Remember, if any thermally accessible conformation of a molecule is achiral, then the molecule is achiral.
Sometimes it helps to build a model or use a Newman projection to help with the stereochemical analysis. Here is such a projection for your chloro-catenane. Note that it still possesses a plane of symmetry (a plane perpendicular to the screen).
If we go further and add another chlorine atom, but now to the back ring, then the molecule becomes chiral — just like 1,3-dichloroallene.
The compounds attached seem to be enantiomers. I don't see how by rotation they could be super-imposable.
Look at the bottom figure in my second drawing. I've taken your interlocking 5- and 6-membered catenane and rotated the 6-membered ring a bit further along (a conformational isomer; remember, as I mentioned above, if a molecule, or any conformation of a molecule, contains a plane of symmetry, then the molecule is not chiral [achiral]). The plane of the screen now contains the 5-membered ring and it bisects the cyclohexane ring. The plane of the screen also contains the methyl group. The plane of the screen is a symmetry element of this molecule; therefore this molecule is achiral.