Yes, it is possible.
Bybit. You’ll need two exchange accounts for that. One will be linked to your strategy, and another one can be used for copy trading. You cannot do both things on one account.
Binance. You can be both trader and subscriber with one account if you submit a Futures strategy and follow a Spot strategy or vice versa. Exchange forbids creating multi accounts.
Binance, Bybit and OKX. It is also possible to use subaccounts for these exchanges. You can subscribe to a strategy and trade with your own strategy if you use a subaccount. You can create up to 5 subaccounts on one OKX account.