There are many different communication protocols used for radio controlled models and even for the same vendor the parts/versions are not always compatible. While most use the same chips to build their systems everybody uses a different protocol and so the different parts do not understand each other.
Most of the companies do not release official information how the protocols works and are not interested to make their products compatible with a competing vendor.
So if a drone claims to be DSM2 compatible there are two options: a genuine spektrum receiver is used or somebody found out how the protocol is working by reverse engineering the system.
Most cheap drones do not use genuine receivers since the product costs less than the receiver alone. The drone you linked does not state DSM2 compatiblity so I'd say the problem is that you have no compatible tx. But information on the product page is not sufficient to tell what they are using.
You could try to find out what protocol the drone uses and check if the multiprotocol project supports it: https://github.com/pascallanger/DIY-Multiprotocol-TX-Module/blob/master/Protocols_Details.md
But even in that case you would need to buy a multiprotocol module and make sure it works with your DX5e.