The ecash and the sats within both systems are NOT the same.

In cashu, the ecash tokens are just that, tokens. New cryptologic proofs which have no relation to Bitcoin or sats. Lightning MAY be used, but is not required, to bridge to other nodes and participate in existing networks. A token from mint A labeled as "sats" is not directly compatible with the same token from mint B. Both mints are doing a swap when sending value from one to the other. When you send sats to a mint, you're effectively selling your sats for an arbitrary token. When you want to use your tokens in a mint to pay a lightning invoice, your hoping the mint fulfills that request at a reasonable exchange rate.