https://www.receive-sms-online.info/ doesn't use fake numbers. The number is valid, and can in fact receive SMS. And that is your entire problem.
They are valid numbers within the national number plan. They are functioning numbers. There is probably no property that is easily visible to you that announces that text messages sent to one of those numbers is publicly accessible.
If you have to verify the identify of the person in front of the computer, you need an entirely different framework. Cell phones work because it's slightly harder (but not hard) to get a new number if you need multiple accounts.
In for instance the UK, it takes maximum of ten minutes and a couple of pounds to get a new, anonymous sim card. I habitually have tens of sim cards floating around with perfectly valid numbers, yet you'd be hard pressed to link any of them to me.
In short: use verification SMS to make it slightly harder for users to sign up with multiple accounts and provide fallback communication. You can't use it to totally stop people from having multiple accounts.