Does Mel and Ambessa Legends trigger on the same chain you empower a unit? For example, I empower a unit, it goes on the stack, and then Mel Legend triggers, that goes on the stack and will resolve first before the unit's empower trigger?

No — it's the opposite order. The unit's Empower resolves first; Mel (Soul's Reflection) and Ambessa (Matriarch of War) only trigger after that, on a fresh chain.

"Empower this" is the effect of the activated ability — it happens when the Empower resolves, not when you activate it (Rule 827.2, FAQ #11993). The legends' "When you empower something else" condition is evaluated only after that inciting event is processed (Rule 383.2.c), so they can't join the chain on top of the Empower. The unit becomes Empowered, then both legend triggers fire simultaneously and go on the new chain — you choose their order (Rule 383.3.d), and they resolve one at a time.

One edge case: if a reaction removes the unit before the Empower resolves, the Empower does nothing — no empower event occurs, and neither legend triggers. Also note "something else" means empowering the legend itself doesn't trigger that legend's own ability, though the other legend would still trigger.