Some background information about options to facilitate the example in my answer. There are 6 basic synthetic positions relating to combinations of put options, call options and their underlying stock in accordance to the Synthetic Triangle. They are all variations of:
Stock + Put = Call
which is the core of put/call parity (details not important here). This equation can be factored 3 ways and since you can be long or you can be short, that makes 6 outcomes.
Synthetic Long Stock = Long Call + Short Put
which means that if you buy a call and sell a put with the same strike price and the same expiration, you have a combination option position that mimics the behavior of long stock or as you called it, a replicating position. This combo has the almost the same P&L as owning the stock.
The cost of this position would effectively be the risk free interest rate, ignoring dividends. Dividends lower a call's premium while increasing a put's premium so the larger the dividend, the less the synthetic position would cost... and the combo could possibly be done for a small credit if the dividend was large enough. For all intents and purposes, let's just pretend that the cost of the synthetic long is zero.
If long term options were used to create the combo, the P&L of the two positions (combo versus stock) would be approximately the same at all times. Option volatility would tend to have minimal effect since premium change would offset (you are long one option and short the other). In comparison, the cost of the respective positions would be vastly different. Therefore, they would NOT cost they same today and they would both NOT be worth the same amount a year for today. However, their P&L would be the same.
A hedge is any non correlated position that reduces the risk of an adverse price movement in your asset, albeit in varying degrees. Some examples of hedging long stock are covered calls, long puts, option collars, a short position in another stock, inverse securities, etc.
Regarding your question, if the replicating portfolio was truly 1:1, then being long the stock and short the replicating portfolio would be a hedge but since they are respectively plus and minus 100 delta, they would offset each other at 100% and there would be no profit or loss potential while both positions were on. Or as you stated: "Buying and selling it just corresponds to buying and selling the stock: but I already own the stock, so what do I need the replicating portfolio for?" The only reasonable answer I can offer is that usage of this might be to defend a greatly appreciated stock during a market decline where you don't want to sell and incur a large tax bite. The size of the correction would determine the efficacy of this.
Where this gets beyond the scope of this Q&A is that there are some arbitrage possibilities for natural and the replicating portfolio but that's much higher on the food chain than we are (professional traders, investment banks, etc.).