First of all, arthropods are animals that are invertebrates and have a tough exoskeleton made out of chitin, a tough, sugary substance. Lobsters, crawfish, shrimp, insects, and spiders all fall into the category of being arthropods. Invertebrates are animals without a backbone. The exoskeleton gives protection and support in the absence of an internal skeleton.
Fungi are members of their own categorization. They have characteristics of both animal and plant cells. The characteristic they share with arthropods is they have chitin as part of their chemical makeup as well, primarily in their cell walls, which serve a function similar to the exoskeleton in arthropods. The cell wall gives the fungi the strength and support it needs to hold itself erect from the ground.
So the shared characteristic is they both have chitin as part of their chemical makeup. The difference is the structure the chitin is used in, the exoskeleton for the arthropod, the cell wall for the fungi.