In my limited experience, I cannot think of any mathematical concept which is not obviously linked to the intuitions we have about the real world (irrespective of whether these are actually true or not).
Numbers can be used to represent physical quantities, such as heights and weights, or just the number of physical objects, such as the number of rooms in a house etc.
Similarly, sets can be used to represent collections of things, such as the collection of all elementary particles in the universe etc.
However, the concepts of number and set are very basic and very concrete concepts, and therefore easily traceable to the real world, but mathematics seems to be constantly evolving from concreteness towards abstraction, from Pythagoras's numbers and Euclid's geometric figures, towards the use of axiomatic systems involving more and more abstract concepts.
So, given this tendency towards more abstract concepts, are there mathematical concepts which we are unable to think of as meaningful representations of real-world things?