the concept of paradox is not a mathematical concept. The meaning of the word paradox is something that exhibits a strong counter intuitive behaviour. Something that looks ok but upon closer inspection reveals itself to be highly problematic, to the point that it is totally hopeless.
Russell's paradox is a seemingly legit definition of a set, i.e., the set of all sets that do not contain themselves as elements, which upon closer inspection leads to an unrepairable situation; no answer to the question "does that set contain itself as an element?" is consistent, and thus the question is unresolved.
The paradox you describe above is similarly a seemingly legit definition of a set, i.e., the set of all sets that do contain themselves as elements, which upon closer inspection reveals to be problematic; the question "does that set contain itself as an element?" does not seem to be answerable from the definition of the set, and thus the question is unresolved.
Notice however that the paradox you describe is weaker than Russell's. Russell's paradox leads to a contradiction. The situation you describe is not a contradiction. It just shows that no answer to the question immediately leads to a contradiction, and thus none can be immediately discarded. However, it leaves open the possibility that a more clever argument can resolve the question. Of course, you will have to carefully axiomatize your paradox in order to really answer such a question.