Wordsworth used the phrase "organic sensibility" to refer to natural mental capacities. Wordsworth writes, "an accurate taste in poetry, and in all other arts...is an acquired taste, which can only be produced by severe thought, and a long continued intercourse with the best models of composition." So he seems to say that, while writing poetry might come naturally, reading and understanding it doesn't. He also is implying that most good poetry is beyond the comprehension of those without a background in how to read it, and that people of less discriminating taste can't be sensible of which poetry is good and which is bad.
He's describing something profound, and in using the word sense, he seems to invoke many of sensibility's meanings: perception, emotional consciousness, sensitiveness. But this use also seems to broaden the definition of "emotional connection to works of literature, music and art" to include nature.