I would have to disagree with Zachary. He's right that, for example, if someone says "I love you" then you would never respond "I agree." However, it would make sense to respond "I feel the same way". And at the same time, "I feel the same as you/I feel the same way" can mean exactly the same thing as "I agree", depending on context. This is an interesting question since the different phrases have different meanings in different contexts, but can sometimes mean the same thing.

The first two are the same, but that last one doesn't fit. We don't usually use the word "agree" when we talk about feelings. "Agree" is usually used when taking a similar position on a subject as someone else.