Thanks for the additional details, Mahtu. Here goes:
To dispose of a surface issue first, the transformation of a panther to a girl is the main plot of
Cat People.
http://www.imdb.com/title/tt0083722/http://www.imdb.com/title/tt0034587/Even if you haven't seen the movie(s), this is part of the culture. So, there is a good chance that you simply borrowed that particular image and worked it into your dream where it fit.
To dispose of another issue, I do not do dream interpretation. Sometimes, however, it is possible to recognize motifs in dreams which many other people have also dreamt. If so, then it might be interesting to consider what the motif seems to have meant to many of those other people. However, only you can decide whether your dream means something similar to you.
Both "wild" animals and opposite sex-same age strangers are frequently encountered symbols for aspects of the source of dreams itself (or call it the subconscious, unconscious, objective self, neglected self, ... ). So a wild animal who is a girl, an identity with cultural resonance, sounds especially promising as a symbol of the dreamsource.
Similarly, a house often sees symbolic service as the whole mind. The source of dreams might have its own space within the house. Sometimes that's a basement, or an attic, but a cage in the backyard will do nicely, and fits in nicely with the animal symbolism.
A cage is a peculiar enclosure in that it is open to the world even if it is confining. The big glass back door has a similar property: a barrier that nevertheless lets the outside see in and in the inside see out.
You are suprised that your source is friendly, but you shouldn't be, since she is just as much you as you are. The difference is that she knows it. You are inhibiting her free movement, and keeping her out of your life, when she would prefer that you work together. The cage was your idea, not hers, so to speak.
No story without a plot, and so we have what
seems to be a little exploration of why she's in a cage. That is, why you put here there.
She does 'have a mind of her own' and will have a different take on things than you do. There is that difference of opinion about the neighbor's dog. (Odds are he's an aspect of you, too, but a tame male canine foil to a wild female feline borders on symbolic overkill.) So, since you can never be sure what she might do, it's a little dangerous to leave her wandering about - although that isn't what she seems to be proposing.
For example, when your significant other asks you whether those slacks make her look fat, Lady Panther would say yes, if that were the truth. You know better than to say that, especially if it is the truth. You're civilized, and she is at best more tame than she might appear.
Animal control, indeed. In real life, how civilization deals with whatever bit of the wild might wander into well ordered areas. In the dream, yet another aspect of you.
And by now you are about ready to wake up. So the dreamsource takes one last shot at grabbing your attention, to ensure that you will remember this dream after you wake up.
And she is a shamelessly calculating pussycat, and so she hauls out her best aspect, and the one that you as a psychologist have run across, Jungian or not (you probably are not his brand of therapist, but you sure are his kind of dreamer), your
anima...
Your age because she is your fraternal twin (so to speak), born when you were, and not your sex because she is your complement.
... being carted away in a paddy wagon, rounding out the cage motif.
Damn, she's good, even if
Cat People did that one first. And she has gotcha.
As I say, that works for you or it doesn't. If not, that's cool. But you more than most people here can check out the bit about other people using these symbols within common motifs, and that's all I can put on the table.