The character pictures a knife
刂 splitting something into a separate piece
另. Every modern sense radiates from that splitting image: cutting apart → distinguishing → 'other' (the separated one) → 'don't' (cut it off, stop doing that) → parting from someone (separating). The fourth-tone
别 reading in
别扭 comes from a different ancestor character
彆 — same simplified shape, unrelated meaning ('stubborn / contrary').