Yes you're talking about an off-chain solution. Exchanges use this solution.
It would be prohibitively expensive and impractical to transfer ownership on-chain for small amounts and/or frequent changes, so you send your coins to a single wallet owned by the exchange.
The exchange has their own system to keep track of who owns what. This system is private and not transparent, but fast and potentially free. Then when you choose to send it from the exchange to your wallet, a network fee is incurred at that point.
The off-chain solution could also be a block chainblockchain solution, though not distributed as no one else would have any interest in maintaining your businesses ledger.
e.g. with coinbase.com
The Deposit:
Your wallet -> "your" coinbase wallet
occurs on main net blockchain
network fees incurred
technically deposited into a single main net wallet
wallet controlled by coinbase
you trust them with your coins at this point
Interwallet transfer on coinbase:
"your" coinbase wallet -> another wallet on coinbase of "your"s
occurs off the chain
as such it's free
as such is opaque
could be implemented however coinbase sees fit:
vb running in an excelsheet for all I know!
Withdrawal:
"your" coinbase wallet -> Your wallet
occurs on main net blockchain
coinbases single wallet is paying you
network fees incurred
grouped with other withdrawals for network fee efficiency
coins in your full control again