Is the set of languages recognized by a Turing machine with an oracle countable?
Every language can be accepted by a Turing machine with an appropriate oracle, for example an oracle for the very language you want to accept. So if you understand "languages recognized by a Turing machine with an oracle" in this way, then certainly their number is uncountable.

Another way to understand this phrase is the you fix the oracle and then ask how many languages are recognized by a Turing machine with access to this specific oracle. In this case the number of languages is countable, since there are countably many ways to specify a Turing machine.