Two scenarios:

You’ve misplaced or forgotten the tracking number, in which case, the tracking number can be provided to you by the person sending it.

In certain countries, you might be able to pick up the money without the MTCN, but you would have to call WesternUnion.

Source: Pick Up Money – Receive a money transfer at Money Services

The tracking number is the de facto method of how the transfer is addressed and looked up in the system. However, WU does not function without. You, as a receiver, however, can, you would just need to present the appropriate details of the sender and the receiver to verify and match your details against the original transfer.