This is already possible.
It is called an OP_RETURN output in the transaction. If any wallet creator wants to store a description, they can add such a feature to the wallet software.
Note that, storing this extra information would make the transaction larger and would therefore require more transaction fees for confirmation.
It is rarely used for such purposes, and there aren't any popular wallets that have implemented any method for displaying any of this data.
Clearly it is not a popular desired feature.