Assuming that your php installation does work for your websites (i.e. you do have a mysql connection there), the most common cause is that the CLI version of PHP uses a different php.ini, or you have multiple PHP installations and oil uses the wrong one. We see this with MAMP on Mac as well.
So check which php-cli binary is used when you use oil, and check if that binary is the same version and uses the same php.ini as apache does.
Because the message is very clear: the mysql extension hasn't been loaded.
I'm not familiar with Windows so I can't be more specific, sorry.
and is that the same one as you apache install is using?
if you do 'php -m', does it list the mysql and mysqli modules? And PDO if you want to use the pdo drivers?
Ok. So PDO is loaded, but mysql/mysqli isn't.
So you'll have to figure out why it is when you run php through apache, but not from the commandline. Perhaps this is something specific of your xAMP stack, maybe you can ask the question there?
And it isn't an environment issue, where your application uses PDO, but oil is trying to use mysql? Check the db.php config files in the different environments. oil will use development if you haven't selected one.