You'd need a matching receiver to be able to receive the signal. You'll need to decode the signal to capture the data in the signal, and record that data. Then, you'll need a matching transmitter to transmit the signal. Finally, you'll need to read the stored data and reconstruct the data stream to be transmitted, and transmit that.
It won't be as simple as recording the incoming signal like you would record a song off the radio.
But, it is possible. The level of effort, I think, would be reduced if you had an oscilloscope and logic analyzer. How much of the working transmitter owner's product you consume while collecting data will influence the length of time it takes to complete the project, too.