Wow! That is an old handle. 1954 is one year older than me! I'm a little confused, though What is the reason your customer wants to install a handle on the opposite door? Normally the other door in a patio frame is stationary. Is this a different case?
Either way, I don't think we'll be able to find an exact replacement for this. I'd be surprised if it's made anymore. We may be able to find a substitute, but I would need more information. Please confirm the distance between the installation hole centers and the thickness of the door. I would also like a few pictures of this handle from a couple more angles so that I can see more of it. Finally, if you know the name of the manufacturer, that would be a huge help, as well.
Both sides of this door slide. The track on this side is destroyed and is difficult to operate. The other side door slides fine but has no handle only a lock mechanism. I will be back there tomorrow and try to get more details on the manufacturer, etc.