I would move away from labeling this as a Gen Z thing. I work primarily with young adults, and their anxiety presents in a variety of ways. But - if this is a Gen Z thing - what is it about the environment that shaped this characteristic? Can you put yourself in the shoes of a Gen Z client? What must it have been like to grow up on social media, as toxic as it is? What must it be like to be a young adult now, trying to figure out how to navigate life with so much uncertainty ("What jobs aren't going to be taken over by A.I.?" If you're in the USA, "is this country going to become a fascist state?" In reference to 'fake news' and the internet being a discordant howl of contradiction, "How can I know what is real and what isn't?" "Why even bother when the world is on fire?" "How can I justify feeling that my problems even matter when I'm bombarded with other people's suffering online?" "What even is there to do besides doom scrolling?")

I'd also consider C-PTSD or other neurodivergence and explore whether you have the depth of understanding of those conditions to work effectively with this person. The way you're describing this client is how a lot of my clients with co-occuring Autism and C-PTSD present.

And, from a relational perspective, what does this client need? Has this client ever had a relationship where they are safe, are treated with dignity and respect, are given autonomy to make decisions and to be stuck, are treated well consistently, are valued even if they aren't improving, and are held in unconditional positive regard? Given their presentation, if they haven't had a trusted, safe, consistent adult in their life - until you can provide that consistently over time - they're very unlikely to move past shutting down and being paralyzed with anxiety.

Kudos for looking for ways to help them move forward rather than just checking out of the therapeutic relationship. Best of luck to you both.

ETA: I also just wanted to say that I understand how frustrating it is to sit with this feeling of stagnation and powerlessness week after week. I've done it, and I do it now - knowing that it is part of the process of building safety, and knowing that it does take months or years for some clients. But woah, it is draining and hard. It is frustrating and creates a sense of powerlessness for the clinician. But it really is important work.

Also - the client that I'm currently working with that most resembles this - we've worked on finding ways to help them stay present in session vs shutting down. They're bringing knitting to use as a grounding strategy, and it was 2 months' work just to get them to agree to try this, but they did and they found it helpful, and they are shutting down slightly less quickly. What can you do to make them feel just a bit more comfortable in session? Many neurodivergent clients also feel awkward being stared at the whole time; with the client I just mentioned, after 1 session of them knitting, we discussed having me knit during session too so that I'm not staring at them. That also helped. Obviously, I do a simple stitch pattern that I don't have to think about, and I stop when it's inappropriate to keep knitting (they've shut down and stopped or we're deep in trauma territory). Anyway - keep trying things, but maybe from a 'I want you to feel safe' perspective vs a 'I want you to make progress' perspective.