A driving licence in English is the piece of paper – or card – that says that you have qualified as a driver. It is not the test itself. First comes the test, and if you pass it, you get a driving licence.
So if you say, “I am doing my driving licence” it makes no sense. You need to do your driving test. And in English, we ‘take’ tests, rather than do them. So, correct is, “I am taking my driving test.”
One other mistake is to say, “I am passing my test today”. This, in English, sounds incredibly arrogant and over-confident. What the student means is “I am taking my test today.” Of course you hope you will pass it, you just cannot say you are passing it!
So, to clarify:
A: Do you have a driving licence?
B: No, I am taking my driving test tomorrow, so by next week, I hope to have a licence.