How can I get programmatic access to the "Date taken" field of an image or video using powershell?

I can't test it right now (don't have any images with XIF data laying around, but I think this should work:

[reflection.assembly]::LoadWithPartialName("System.Drawing")
$pic = New-Object System.Drawing.Bitmap('C:\PATH\TO\SomePic.jpg')
$bitearr = $pic.GetPropertyItem(36867).Value 
$string = [System.Text.Encoding]::ASCII.GetString($bitearr) 
$DateTime = [datetime]::ParseExact($string,"yyyy:MM:dd HH:mm:ss`0",$Null)
$DateTime

In general, you can access any extended property for a file shown in explorer through the shell GetDetailsOf method. Here's a short example, adapted from another answer:

$file = Get-Item IMG_0386.jpg
$shellObject = New-Object -ComObject Shell.Application
$directoryObject = $shellObject.NameSpace( $file.Directory.FullName )
$fileObject = $directoryObject.ParseName( $file.Name )

$property = 'Date taken'
for(
  $index = 5;
  $directoryObject.GetDetailsOf( $directoryObject.Items, $index ) -ne $property;
  ++$index ) { }

$value = $directoryObject.GetDetailsOf( $fileObject, $index )

However, according to the comments on another question, there is no general-purpose mechanism for setting these properties. The System.Drawing.Bitmap class that EBGreen mentioned will work for images, but I'm afraid I also do not know of a .NET option for video files.