It’s going to depend on how the access is set up. It could be set up such that the only way into that network is via that browser thing.
You can always connect to yourself from the Windows machine and tunnel SSH over that, but it’s likely you’ll hit a firewall or possibly even a TLS MitM box.
Virtual desktops like that are usually used for security, it would be way cheaper and easier to just VPN your workstation in. Everything about this feels like a regulated or certified secure environment like payment processing/bank/government stuff.
They could also just be spawning Windows VMs directly in AWS, no point doing nested virtualization for something like this. Pretty sure they have a service for doing exactly what you described. No need for a VPN, it can spawn your VM on the right network already (they call it VPC). They can even put real GPUs for AutoCAD and stuff on those things.