⺼ is the flesh-meat radical (visually a "moon" but historically 肉) and places 肤 squarely in the body-parts family - same radical in 胸, 脏, 肝, 脸. Skin is the body's flesh envelope; the link is direct.
夫 gives the sound fū exactly, same tone. The simplified 肤 borrows 夫 as a clean phonetic; traditional 膚 has the much more elaborate 盧 element. No semantic content here - pure sound match.
