Why serial number on my MacBook Air shows invalid?

To add a bit to the other replies, an "Invalid Serial Number" message generally indicates that the device you have was serviced or even replaced prior to your ownership.

If you purchased this device new and have never had it serviced or replaced, then contact Apple Support and they can correct this error for you. Just be aware that they may require that you provide a copy of your purchase receipt to show that you're the original owner/purchaser.

However, if you purchased this device second hand, then it's likely that the original owner had the device replaced but did not return this device to Apple as they should have. If this is the case, they should not have sold the device and you should make an attempt to return it for a refund.

Try copying & pasting the serial number from the System Profiler (aka System Information). Option-click the Apple menu & select the top option. Sometimes extra hidden characters can be included when copying from some locations. Usually copying from "About This Mac" should work, but you never know.

You can also try typing the serial number into the field, but you must be very careful to distinguish between some numbers & letters since they can appear like a different character or number.

Double-check the serial number matches the one etched into the Bottom Case (you will likely need a good light & magnifier to read it).

You might need to confirm it with apple. Have valid proof of purchase information ready and Contact - Official Apple Support to have it confirmed

Was it purchased from a non Apple Premium Reseller? If thats the case, you need to contact Contact - Official Apple Support, and have the valid receipt/ proof of payment to be the basis for verifying the Macs purchase date.