1. This is probably the first winter of my life that I'm really enjoying.
- This sentence is correct. This one says what you're trying to say.
2. This is probably the first winter of my life that I'm really being enjoyed.
- This second sentence, however, doesn't much sense. If you say that "I'm really being enjoyed" it sounds like someone else is enjoying you. For example, in the sentence "the cookies are being enjoyed by all the guests", it means that the guests are enjoying the cookies. The second sentence doesn't say that you are enjoying the winter, but that someone is enjoying you this winter.
I hope that helps!