Here is a quick PowerShell Script which helps you to eassily count your SQL Databases and show you the result in a simple manner.
$sqlpassword = "PASSWORD" | ConvertTo-SecureString -asPlainText -Force
$sqlusername = "USERNAME"
$sqlcredential = New-Object System.Management.Automation.PSCredential($sqlusername,$sqlpassword)
$sqllist = Get-SqlDatabase -Credential $sqlcredential -ServerInstance INSTANCE NAME
$sqllist.count
Enjoy!
