This is ultimately derived from Kant, who taught that our senses are unreliable. This led later philosophers to dispense with external reality and start with sense perception.
This makes a certain kind of sense. If you do not believe that your senses observe reality as it really is, then why not dispense with external reality and start with sense perceptions? This is essentially what Kant followers, including Bohr, did.
Let’s now look at Bohr’s acceptance of the particle-wave duality that comes up when identifying the nature of light and hence his rejection of the Law of Identity.
Rejection of the Law of Identity
“The situation which we meet here is characterized by the fact that we are apparently forced to choose between two mutually contradictory conceptions of the propagation of light. One, the idea of light waves, the other, the corpuscular view of the of the theory of light quanta, each conception expressing fundamental aspects of our experience. As we shall see in the following, this apparent dilemma marks a particular limitation of our forms of perception which is bound up with the quantum of action.” 
We have already mentioned this particular quote when discussing Bohr’s acceptance of contradictions and the alleged failure of our forms of perceptions and his rejection of reality and objective facts.
But now, let’s take a look at the particular aspect of this quote that points to his belief on the nature of light.
Bohr accepts the contradiction that exists between the wave theory of light and the corpuscular or particle theory of light. He accepts this contradiction, taking it as an indication of the limitation of our faculty of perception and in doing so rejects the law of identity rather than reject the contradiction of particle-wave duality and accept the law of identity.
Is particle-wave duality a reasonable concept and could it be possible that this is a valid part of quantum mechanics? No, even a casual grasp of the Law of Identity makes it clear that particle-wave duality is an impossibility.
Everything that exists has a specific nature. Or, to put it simply: To be, is to be something. Everything that exists is subject to this basic principle, be it a mountain or the smallest subatomic particle.
To exist is to have a specific, non-contradictory nature. If something is a particle, it is not also a non-particle.
The concept of “particle” and “wave” are mutually exclusive. A particle is a kind of physical entity. It is one form physical entities exist in. It is simply one way of saying “that thing that exists, it is this kind of thing”.
But the concept of a wave is not the same kind of concept. A wave is not a form of physical existence. It is an abstraction which identifies relationships. It identifies relationships between the properties of entities, which might be physical entities or other abstractions.
For instance, “water waves” do not have a physical existence. The “wave” does not have physical existence.
What physically exists is water. What we refer to when we point to a “wave” of water is water arranged in a certain kind of spatial relationship. That is a certain kind of relationship between the position of water molecules.