how does the software look like?
It is already there, and is implemented in centralized exchanges.
but how the sofware will operate is not clear to me
It is quite simple. The bank has some public key, which is displayed to the user. You deposit BTCs there, and they appear on your account, exactly as when you deposit physical cash into some ATM, and it is credited into your bank account.
issuing their own digital cash currency
This step is optional, and it happens only after some time. First, you have just deposits and withdrawals, and you have no token, owned by the exchange. However, as things are evolving, it is profitable to release a new token, and then you have for example things like BNB on Binance.
Some would be fractional reserve while others may be 100% Bitcoin backed.
If you have just some web page, with "deposit" and "withdraw" buttons, displaying you just some amounts, then you can have any kind of fractional reserve. But: when some exchange is getting more serious, then it is slowly moving into "100% Bitcoin backed", just to convince users, that it wouldn't collapse, as other exchanges in the past.
Was the Lighting Network being described here, or did he talk of something else?
Would this look like in practice as a sidechain?
None of them. Because when you have LN or a sidechain, then you can explore many details, related to your coins. So, it no longer behaves like a typical bank. Because if you deposit some cash into some ATM, then can you really go into your bank account, and trace your banknotes, by seeing their serial numbers? Can you see, that your $100 bill went from bank A to bank B, then it was taken out of another ATM, and deposited back into bank C? Can you count all coins, and audit the supply?
In case of LN and sidechains, you can see a lot of traces. In case of exchanges, you just see, that you have N coins, you can see some order book, and check the latest price. But: all of that is not backed by any proofs, exactly like in banks, so after you deposit 1 BTC, you no longer know, if that exchange still have your 1 BTC or not. You just trade, and things are getting "real", and you can get some "details", only when you click "withdraw" (while in sidechains and LN, you need to approve your actions with your signatures, and you can read a lot of traffic, related to the whole network).