Differentiate among ASA of the same type

Is it possible to create a non-fungible ASA, e.g. MY_STUFF with assetid x, and distinguish among token of the same type, e.g. MY_FORK, MY_CUP, …, all being of MY_STUFF type and assetid x?

Do I have to create multiple ASAs or can I achieve this with just one ASA?

To create non-fungible assets, you need to create one ASA per non-fungible asset. Each ASA / non-fungible asset will have a different asset ID.