I think the author is saying that the heritability of a trait depends on the range of environments being considered, but saying it in a needlessly grandstanding and obfuscatory manner?

Having now read the paper, yes, that was what he was saying, and the non-quoted sections of the paper are much clearer. Example

For the modeling underlying twin studies to make sense, one has to assume that the environments experienced by identical twins raised in the same family are no more similar than those experienced by frater- nal twins, that genes and environments are independent and additive (again, in the usual ANOVA sense) in the determination of the trait, and that parents mate randomly with respect to the trait.

That's totally true. If you compared an organism with a randomly generated genome to a human and exposed both to what would be a normal range of environments for a human, you would find that all traits are entirely genetic. And if you selected people from the normal range of human genetics but only gave half of them access to important things like a uterus then you would find that all traits are almost entirely environmentally determined. So you can't say that "Trait Foo is 50% heritable" unless you also specify the populations and range of environments you're talking about.

EDIT: Or you could say that heritability is ordinal but not cardinal, given some reasonable assumptions.