Your wife is probably a big part of why your SD doesn't respect you. It's fine if she wants to do the major disciplining but you are an adult in your SD's life and your wife should be teaching her daughter to respect adults, period. My SD would have sat in the middle of the street if I had told her to.

Is there an actual diagnoses that is the cause of the anxiety being difficult?

There has been some same sex couples on here, not sure about currently. While we all understand step related issues, I'm sure your situation can be even more challenging. Do you think your SD has issues with her mom being a lesbian? I just ask because she doesn't seem to have a problem with her bio dad's wife.

I hear ya on the whining and attitude. My SD was not that way when she was young, but my niece sure was (and still can be, at 17).

It is unfair to ask someone to parent yet not allow them to discipline. They both go hand and hand. Has your SD been disagnosed with anxiety or is this an assumption being made ?

But that never made me a parent to her prior kids. I really couldn't articulate that properly until I found step talk. Her offspring have two parents and I'm not one of them became my mantra. Disengagement saved my marbles!  If I'd have know how painful so many of those early years of us being together were going to be - I would have run.  Fourteen years later it's ok - she knows I will never live under the same roof as either of them.  I also don't believe gender or sexuality has much to do with these kinds of dynamics. 

I'm bisexual, but married to a man. I went on a few dates with a woman years and years ago that had a kid, and the BF was not thrilled about it (so I heard). But we weren't "together" so I didn't do anything with her kid; only heard from her that the father had "issues" with her being with women and men post-separation (personal theory: it's because her sexuality didn't benefit him while they were together, and he seemed like a jealous prick based on her description).

So, my stepparenting experience isn't from a same sex relationship lens, but based on the other same sex relationships that have been presented here, RARELY are the issues and solutions different because of that fact. Here's a likely list of causes for this behavior:

1.) Your wife isn't really disciplining her daughter. "Talking to" is not a form of punishment, but can be a form of reform. I was the kind of kid who was crushed if my parents told me they were disappointed in me and would correct my behavior immediately if they talked to me about it. However, even with being a kid who "talking to" worked with, I still got punished when I misbehaved. I'd lose going out with friends or doing an activity, or I'd sit in time out. 

There still has to be a consequence for the bad behavior to solidify that it's bad. Being "talked to" rarely works. If it worked, employers wouldn't create systems of consequence for adults to not eff off on the job (that's a more complex issue, sure, but as a general concept it works). The threat of punishment or the promise of reward tied to behavior is what usually gets optimal behavior.

What "talking to" only does is erase that system. The punishment is having to sit still for 5 minutes while you get yelled at. The reward is getting to go right back to what you were doing for hours on end with no end in sight of it stopping except for another 5 minutes yell fest sprinkled in.

If that is your DW's way of disciplining her daughter (and it's very common in step families, which is why I mention it), then SD is going to be a holy terror if disappointing her mom isn't enough to make her want to behave.

2.) SD has an actual mental health issue, and your DW and her XH need to get SD into treatment. Going through life without proper treatment for mental health hurts big time. If she is having meltdowns whenever she does something wrong, she needs a therapist to work through what is causing that. Which somewhat leads into my next point...

3.) Her dad plays a big part in her life, and what happens in his home can impact what happens in your home. On the positive side, BF may run a very tight ship, with clearly-defined rules and consequences/rewards, and requires SD to respect her SM (and probably also you and your DW). BF may also give SM the authority of a full parent in his home. Those things will make SD behave differently. Kids will behave based on how they're allowed to behave most of the time, so a tight ship is going to produce a respectful kid (usually, there are always exceptions and deeper dives into how to parent towards a kid's currency, but I digress).

On the negative side, it's possible that BF and SM's house have no rules, so SD bucks at what your DW puts in place as rules and consequences. Or, BF is actively trying to turn SD against her mother (parental alienation), so he lies about DW to SD, which pushes SD to misbehave to appease her dad. Or, BF doesn't respect your DW's sexuality and doesn't see your marriage as "real" and has rubbed that mentality off onto SD. These things are difficult to combat and require a lot of work on your DW's part to correct, up to going back to court to limit time between SD and BF (which would be a very, very, VERY last resort).

4.) Your own behavior may push SD away and cause DW to tell you to butt out. We see many SPs on here who get railroaded by their partner when trying to get SKs to just behave civily. However, we do sometimes see SPs who take it too far or have very bad to abusive ideas about parenting that make complete sense why their partner would tell them to stop. It can also be SPs who get upset at totally normal kid things, which is an issue with the SP, not the kid.

This is by no means a comprehensive list of things that could be happening. Hopefully it gives you a place to start to examine your own situation and perhaps find some areas that may be of concern. Ultimately, your wife said to let her handle it, so let her handle it. Tell her every time SD does something (or doesn't do something she is told), and tell her every time SD is disrespectful to you. Perhaps being faced with everything going on, DW will ask for assistance (or you'll realize that your DW doesn't actually care about your comfort in your own home, which happens quite often in step families).

We can help you will solutions as you figure out more of what the problem is. Best of luck.

Nope, not me.  But, my bride and I are navigating being the parents of a gay man.  Our son (my former SS-27 adopted at his request when he was 22) came out when he was 20 to his mom and I.  We three became a family when  his mom and I married the week before SS turned 2yo.  His mom made me his dad when we married and we raised him together.  Neither of us would have abdicated such a major part of our life together.

Be wary of a partner who will not prioritize you not only as their partner but as an equity parent to any children in  your blended family.

Regardless of the demographic of the partners, if the two of you are not both equity life partners and equity parents to any kids in the mix regardless of biology the odds of a successful life long relastionship are poor at best.  

Do not tolerate your SO isolating you from any part of the life you share.  If she does it and you tolerate it, cut your losses now and move on to a partner without baggage who can share her entire life with you, and you with her.

Just my thoughts of course.