My solution was:

First, we can see that a maximum of 2 songs can be liked by any pair of girls at the same time, because if 3 songs are liked by, say, both A and B, then J cannot like any of these (otherwise it would be liked by all 3 girls). So J likes only one song. But she must share this song with A and with B, again a contradiction.

Now consider the case where only one song is shared for each pair. The first pair has 4 possible songs. Then the second pair has 3 possible songs (since the first song cannot be liked also by the second pair). Then the third pair has 2 possible songs (again for this reason). Finally, the last song will be either liked by A, B, J or nobody at all (but not two people at once, because I consider the case where only one song is shared in each pair). This makes 4 additional possibilities. In total, there are 4*3*2*4 possibilities in this case.

Now consider the case where at least one pair shares two songs. Only one pair can share two songs together, otherwise, say A and B like songs 1 and 2, and B and J like songs 3 and 4. Then the pair A and J will like either 1, 2, 3 or 4 but then this song will be liked by everyone, which is impossible. So there is only one pair sharing two songs. There are 3 pairs, so 3 possibilities for the pair that shares two songs. Now for this pair, there are 4 possibilities for the 1st song, and 3 possibilities for the second song. The second pair has two possibilities for their own song, which leaves only one possibility for the third pair. In total, there are 3*4*3*2*1 possibilities in this case.

These are all the possible cases, so adding everything we get 4*3*2*4 + 3*4*3*2*1 = 96 + 72 = 168.

So there are 168 ways possible.