bash is more akin to cmd/batch then Powershell.

there is a TON more you can do with powershell than you can with bash. Plus object pipelines are superior to text pipelines

You can also install Powershell on Linux. So I would say Powershell is the Powershell of Linux

I don't think powershell is even the powershell of windows. You get some of the better sysadmins using it, but working in the c#/windows world I see developers rarely touching powershell, certainly nowhere near as often as you'll see Linux develops writing a shell script. I wouldn't be surprised if C# was more used for "scripting" purposes than powershell, it usually ends up shorter and more readable anyway.

you can install bash on windows too.