The phrase "contact number" refers to a phone number that you can call or message someone on. It's a way to easily reach out to someone and communicate with them using their phone. Full definition
Certification and licenses: Name of license or certificates you gained, date and address, name as well as contact number of the institution where you get the license or certificate. (resume2016.com)
You should also carry emergency contact numbers for your pet in your wallet. (moneygeek.com)
If a cat is not yet fully recovered by the morning after surgery, call the emergency contact number provided on the medical record provided at the time of discharge. (ocgainesville.org)