How can I add a release date of a podcast to the filename or file metadata?

Looks like gPodder already has a built-in extension which does this. No need to download it as it's included.

1. Click gPodder > Preferences.
2. Click the Extensions tab.
3. Check the box next to Rename episodes after download.
4. Click the Edit config button on the bottom-right of the preferences window.
5. Check the box next to extensions.rename_download.add_sortdate.
6. Close remaining windows.

All future-downloaded podcast filenames should then be prefixed with "YYYY-MM-DD -".

Optionally, you can also enable extensions.rename_download.add_podcast_title to prepend the title to the filename too.

You could also use the following PowerShell to rename existing files:

#
# PrependDateToPodcasts.ps1
#
Param (
  [Parameter(Mandatory=$true, Position=0)][string]$FeedURL,
  [Parameter(Mandatory=$true, Position=1)][string]$Directory,
  [Parameter()][switch]$DryRun = $false
)

$XML = Invoke-WebRequest $FeedURL
$Podcasts = ($XML | Select-Xml "rss/channel/item").Node

$Podcasts | ForEach-Object {
  $Filename = Split-Path $_.enclosure.url -Leaf
  $Date = [datetime](($_.pubDate -replace " [A-Z]{3}$", "") -replace "^w{3}", "") | Get-Date -Format "yyyy-MM-dd"

  $FilePath = Join-Path $Directory $Filename
  $FilenameNew = "${Date} - ${Filename}"

  if (Test-Path $FilePath) {
    Write-Host "Renaming `"${Filename}`" to `"${FilenameNew}`"."
    if (!$DryRun) {
      Rename-Item $FilePath $FilenameNew
    }
  }
}

# Preview changes before actually running.
.PrependDateToPodcasts.ps1 "http://files.libertyfund.org/econtalk/EconTalk.xml" "C:pathtogPodderDownloadsEconTalk" -DryRun

# Rename for real.
.PrependDateToPodcasts.ps1 "http://files.libertyfund.org/econtalk/EconTalk.xml" "C:pathtogPodderDownloadsEconTalk"