Is there a difference between the phrases "I am fine with it" and "it is fine with me"?
Solution 1:
I see a small difference between I am fine with it and it is fine for me, which is the same difference between I am unclear (used to express doubt or confusion) and it is unclear to me: in one phrase the grammar subject is I, in the other is it. For example, the focus could be placed on I to mean it is unclear to me, but it can be clear to others, or it is fine with me, and it will be fine with the others too; vice versa, the focus could be put on it to mean it is fine with me, but I don't know what others would think.
Most probably, both the sentences (the one starting with I, and the one starting with it) are understood to mean the same thing from most the people.
Looking for the frequency those, and similar phrases, are used, I get the following data (first chart; second chart):
Solution 2:
There is a very slight implication of greater indifference given by "it's fine with me" relative to "I'm fine with it". I don't honestly think either is "on its way out".
Solution 3:
The Corpus of Contemporary American English, which has text from 1990 to 2010, has 9 occurrences of “I am fine with it” (with either “I am” or “I’m”), and 28 occurrences of “It is fine with me”. So, both are still in use, and even though your second sentence seems less used in American English.
Regarding the meaning of the two, I don't think there is any perceived difference.