This is prob a hot take given this is lemmygrad but in my opinion Lenin is rather unfair to Bogdanov in that book. Lenin sees what we directly observe prior to applying any sort of cognitive interpretation to it or even being aware of it, i.e. the raw sensuous experience itself, as a mental thing created by the mind, what he calls a "reflection of reality" as opposed to reality itself. Bogdanov denies that there exists a reality beyond experience / beyond what we observe, and so Lenin interprets this to mean he is saying that all that exists is mental stuff and accuses him repeatedly of being an idealist.

But Bogdanov's main thesis is not that we are trapped in some sort of mental "reflection of reality" and reality and that non-mental reality doesn't exist, but that he is not convinced by the argument that we can't perceive material reality directly. Philosophers often use arguments from dreams or illusions to try and "prove" that what we perceive cannot be directly equivalent to material reality as it actually exists but is sort of a purely mental creation. Bogdanov goes through these arguments and shows how, in his opinion, they don't actually work and can be rather easily refuted.

Bogdanov also is clear that he rejects the thing-in-itself, which Lenin interprets this unfairly as more proof he is an idealist, but Bogdanov is just taking the stance that the material world is not actually divisible up into discrete "things" that can be considered in complete isolation from one another. Everything is interconnected from his view, and it is humans who divide up the world into discrete objects as a way to make sense of it. What is "real" for him is just what we perceive, what we are actually looking at when we label something as a particular thing, like a "dog" or a "cat," and not the label itself.

Bogdanov's view is rather comparable to the later Wittgenstein's views, and there is a modern philosophical school called contextual realism created by the philosopher Jocelyn Benoist that is also rather similar. A lot of idealist philosophers will attack Kant's notion of the "noumena" because if it lies beyond experience then by definition it is not observable and so we can never actually be certain that it is even there or what its properties are, but they will leave the "phenomena" in tact, and thus believe everything is mental. If you read Benoist's book Toward a Contextual Realism, there is a whole category criticizing the notion of phenomenality, criticizing common arguments given, such as those from illusions, as actually demonstrating what we perceive cannot be considered reality as it directly exists.

Bogdanov was not some anti-Marxist idealist, he was a materialist and a Marxist he just didn't agree with metaphysical realism but was a direct realist. Idealism is when you deny Kant's "noumena" but uphold his "phenomena." That wasn't Bogdanov's position, he wasn't an idealist. He rejected the whole phenomena-noumena distinction entirely, seeing them both as wrong, and took a direct realist stance. This is not in conflict with the materialism because the material sciences are driven by what we observe, by empirical evidence, and so if you believe what we observe is equivalent to reality, then the material sciences are a direct description of material reality.