I have done quite a few 6s Plus's and have never had an issue with Touch ID so as far as I know there are no unique quirks to them when it comes to the home button other than the usual (don't damage the ribbon cable, remove battery and drain power before removing connector ect.)
Are you using OEM or aftermarket screen? The home button cable is built in to the screen assembly on the iPhone 6s Series and the aftermarket screens frequently have issues so if your using aftermarket screens you may need to switch to OEM to avoid this issue.
Also make sure the cable is not damaged and is seated correctly into the board connector, if the connector is not quite fully connected then you will get the click function but not the Touch ID.
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Did you checked the ribbon that goes from button to the logic board?
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