He was never a shape-shifter. That is a complete misrepresentation of what regeneration really is - The Zygons were shapeshifters, Mystique from the X-Men is a shapeshifter, so too is Maya from Space 1999, these are characters who alter their shape, appearance, and gender at will and at any time.
With the Doctor and his fellow Timelords the difference (in the classic series that is) is that Regeneration was a failsafe process triggered upon the moment of their death to ensure their resurrection and continued life, albeit in a reconfigured body.
When you look at every Doctor from 1963-96 they all change their bodies while at the point of death, literally they tend to be flat on their backs taking their last breaths when the process kicks in to ensure their ongoing survival. They maintain their gender, and their memories, but their bodies are reconfigured by the process.
What happens with the 2005 revival is that rather than it being his last breath that triggers the Regeneration it seems to be more an act of will, he is always standing when the process is triggered and often it isn't clear in what way exactly he is dying - Peter Capaldi's Doctor is a good example of this, he doesn't appear to be ill or dying. He just... gives up.
But whatever. The Doctor isn't a shapeshifter, not in the sense the modern BBC is selling the concept of the Doctor. It is sad to see all the rules and discipline set up in the classic series have been stripped away by this point in time in the series' history. Any show needs rules, a set of limitations and structures that help to define the character and his world. But we don't have that discipline anymore.
I know that the classic view on him was not a shapeshifter, but since we now have him as a totally unknown alien creature, why not be that he can shapeshift at will, and only regen it due to preconceptions placed in his mind that he was a time lord?