The Copenhagen is a differential cascode circuit. The Copenhagen V uses tubes (valves) for the upper device, while the Copenhagen S uses transistors (sand). They are otherwise just about the same. It does indeed use input and output transformers (amorphous Lundahls in both cases, though the inputs can use mu metal instead). The tubes (E81L's) need not be matched at all.
It is what is often called a "transconductance amp". In essence, it uses the headphone's impedance as an I/V converter, and it takes advantage of the slight Sennheiser impedance bump to make it a bit more euphonic. When used with Senns, I'd describe the sound as happy and punchy (kind of making them sound like how Grados would sound if they were not grating) - some people hate it, others love it, but it definitely has a sound. It can be linearized with the addition of some parallel resistance on the output (this is what a triode amp does except the parallel load is the tube plate impedance). Without doing that, it is not great with the Focals. Even with, they are so poorly damped that it would not be my first choice for powering them. With phones that have a flat impedance, it is pretty neutral. It's not an amp for everyone (definitely not for the objectivist crowd), but it is my favorite these days - I use a Copenhagen S with 650's for most pleasure listening.