How can you check for a specific instance of an App using Powershell/CMD?

You could try using the MainWindowTitle and/or CommandLine properties to differentiate between processes with the same name:

# List the different instances of notepad++
Get-Process -Name notepad++ | Select-Object ProcessName,MainWindowTitle,CommandLine

ProcessName MainWindowTitle        CommandLine
----------- ---------------        -----------
notepad++   C:\A\A.txt - Notepad++ "C:\Program File…"
notepad++   C:\B\B.txt - Notepad++ "C:\Program File…" "C:\B\B.txt"

From there you can write a basic test. For example:

# Check for assertion failure window:
if ( (Get-Process -Name "SDX Test App").MainWindowTitle -like "Assertion Failure*" ) {
  Write-Output "It's running!"
}