@Jakub93 These days, unfortunately, having to deal with customer service is something we all wish to try to avoid. This holds true for both hosts and guests. Their CS reps are outsourced, poorly trained, may be uninformed about correct policy and procedure, give out incorrect information, and are quite frustrating to deal with.
I don't know what the CS rep was talking about regarding a discount. If a guest's stay gets cancelled, the guest gets refunded and Airbnb will sometimes give you an extra $10-$20 towards another place that might be more expensive. Airbnb can't put discounts on a host's listing- the host is in charge of what discounts they offer.
Bummer it happened on your birthday. That's one of the drags about being an adult, isn't it- the rest of the world doesn't take much notice of your special day like when we were kids 🙂