iPhone 8: Home button doesn't move inward when pressed, but iPhone makes a click sound
You need not worry. Since your iPhone is functioning fine, most likely there’s nothing wrong with your iPhone 8.

With the iPhone 7/7 Plus announced in September 2016, Apple introduced a new style of Home button which no longer physically clicks. Similar Home button is built into iPhone 8/8 Plus.

The click that you experience is a form of tactile feedback to let you know that the button press is successfully registered by the iPhone. You need not press it too hard under the assumption that it is damaged and requires a harder press.

You can read more about the new Home button in the Apple support article, How to use the Home button on iPhone 7 and iPhone 8.


  The Home button on your iPhone 7, iPhone 7 Plus, iPhone 8, or iPhone 8 Plus is unlike the Home button on other iOS devices. Instead of moving inward when you press it, the Home button detects the presence and pressure of your finger. When you press the Home button, you feel a slight click.


What you are experiencing is called Haptic Feedback produced by the built-in Taptic Engine. You can read more about the new Home button in iPhone 7/7 Plus/8/8 Plus in the linked articles:


The iPhone 7's new home button isn't a real button



  As well as killing off the headphone jack, Apple has dropped the mechanical home button from the new iPhone 7, replacing it with a static version that's force sensitive. Instead of a button that physically moves, the new home button will use Apple's "taptic engine" to click back at users. Vibrations delivered through the home button will be used as notifications, with unique buzzes for things like text messages and calls.
  
  The new home button, which uses a "taptic engine" to give you physical feedback when you press it — it's pressure sensitive, too, so it can tell if you really mean to press it or just tap it.



Daring Fireball: The iPhones 7



  The home button on both iPhones 7 no longer physically clicks. Instead, it’s a force touch sensor, and it uses the improved Taptic Engine to provide simulated click feedback.


Note: The linked articles are mostly talking about iPhone 7 as it was with this iPhone that the new Home button was introduced for the first time. As far as the Home button is concerned, the contents are equally applicable to iPhone 8.

Additionally, this style of Home button is exclusive to iPhone 7, iPhone 7 Plus, iPhone 8 and iPhone 8 Plus. It hasn't appeared in any of the iPod touch and iPad models so far. The last released models of iPhone have completely phased out the Home button and have instead started incorporating Face ID for authentication.