别 joins 另 (other) with the knife 刂: to cut into separate pieces. Separation spread three ways — parting from people, being 'other/different', and the imperative 'don't' (cut that behavior off!).
别 does triple duty daily: 别客气 (don't be polite — the host's refrain), 别担心 (don't worry), 别的 (other ones: 还有别的吗? anything else?). As parting it aches in poetry — 告别 (bid farewell), 离别 (separation) — a theme Tang poets returned to obsessively, since official postings scattered friends across an empire with no telephones. One knife-cut character: prohibitions, otherness, goodbyes.
Cut 刂 it off from the rest 另 — DON'T; the cut-apart is the OTHER.
别 as 'don't' is literally 不要 (bùyào) fused and shortened — say 不要 fast enough and 别 falls out.