I'm not familiar with msdeploy.exe, but I do seem some errors ...

According to the docs, you should be using a colon (:) after the initial parameter name; such as: -dest:contentPath instead of -dest=contentPath.

Additionally (remember, these are guesses cause I'm not familiar with msdeploy.exe):

1. You're missing the double-quote after TextFile.
2. You shouldn't have double double-quotes around your match value.
3. I'm not sure you want the $ after web.config, but I'll leave it alone since it seems to have found your file.

I think you're final line needs to looks something like this:

& $msd -verb=sync -allowUntrusted -dest:contentPath="$iisPath" -source:contentPath="$iisPath" -setParam:kind="TextFile",match="(MyEntities.*?catalog=).*?(;)",scope=web.config$,value="`$1$newActiveTargetDbName`$2"

I hope this helps!

Update

Based on your working one, which I doubt works since it's using % variable names (not how POSH does things), this should do what you want:

& $msd -verb=sync -allowUntrusted -dest=contentPath="$iisPath" -source=contentPath="$iisPath" -setParam=kind=TextFile,match="(MyEntities.*?catalog=).*?(;)",scope=web.config$,value="`$1$newActiveTargetDbName`$2"

Note: there's also several unanswered variables; if you're giving us the full code then there's a ton of errors.

Update 2

After reviewing everything again (with little feedback from the OP), this is how I would do it in PowerShell. I'm not 100% sure that your arguments are correct:

#region Set These Variables
$computername = $env:ComputerName
$nt_username = 'username'
$nt_password = 'password'
$newActiveTargetDbName = 'newActiveTargetDbName'
#endregion

$msd = "C:\Program Files\IIS\Microsoft Web Deploy V3\msdeploy.exe"
$iisServicePath = 'MyWebsite/MyService'
$iisPath = "${iisServicePath},computerName=${computername},userName=${nt_username},password=${nt_password}"

$start_process = @{
    'FilePath' = $msd;
    'ArgumentList' = @(
        '-verb=sync',
        '-allowUntrusted',
        "-dest=contentPath=""${iisPath}""",
        "-source=contentPath=""${iisPath}""",
        "-setParam=kind=""TextFile,match='(MyEntities.*?catalog=).*?(;)',scope=web.config$,value=`$1${newActiveTargetDbName}`$2"""
    );
    'Wait' = $True
}

$process = Start-Process @start_process

This will give you a command that would look like this in Command Prompt:

"C:\Program Files\IIS\Microsoft Web Deploy V3\msdeploy.exe" -verb=sync -allowUntrusted -dest=contentPath="MyWebsite/MyService,computerName=V-ASUSM32,userName=username,password=password" -source=contentPath="MyWebsite/MyService,computerName=V-ASUSM32,userName=username,password=password" -setParam=kind="TextFile,match='(MyEntities.*?catalog=).*?(;)',scope=web.config$,value=$1newActiveTargetDbName$2"

Note: In the setParam argument, I changed match to single quotes because it looked to me like it was going to mess up your double quotes around kind. If msdeploy.exe doesn't like single quotes, we'll need an actual working command from CMD to figure out what's going on.