I'm not sure which dictionary you used, but it's a little odd that it would use the word being defined in its definition in the way that it did.
In Merriam-Webster, the sense of subscribe that you are referring to is defined as:
3 : to assent to : SUPPORT
This has mostly the same meaning but there is a difference.
Agreed with mostly refers to a single instance of something.
With subscribed to there is an implication of ongoing agreement, if not actual support. This follows the other sense of subscribe where you pay for (or sign up for) something, such as a magazine, and receive regular updates.
So, while you might agree with a vote (a single event), you wouldn't say that you subscribe to the vote.
In your specific example, what's being subscribed to or agreed with is a view, which is something a bit more generalized than a single event. (Somebody can still have the same view months from now.) That's why both words can be used—even if the emphasis is a bit different with each.
So, the following sentence would be understood and not simply taken as repetitious or redundant:
Not only do I agree with that view, but I actively subscribe to it.