It is how English refers to the weather, or to seasons, or to time... it's spring, it's raining, it's five o'clock, it's Friday. These are idiomatic and said all the time, or at least when appropriate or "in season"...
Most people don't go around saying I feel like spring or I feel like Friday or 5 o'clock, etc. This is a simile (a comparison using' like' or 'as') rather than an expression you are going to hear often. It's not something I've ever said, although I love spring!
In any case, "I feel like Spring" definitely does not mean the same thing as "It feels like Spring". The latter means that the world has Spring-like qualities, while the former can be interpreted as something like "I feel like doing things today that are typical things to do in Spring."
