For example, I want to make new booking.com.
I will develop and design the new website from scratch. Functionality will be identical like to that on booking.com. I will not take any logos, images or text from there - just functionality.
Are sites like booking, airbnb etc protected in any way? To me, they are offering tools that are more or less similar to many other websites out there doing similar things.
Under English law, few published websites will benefit from any protection in relation to their functionality, even if that functionality is new and original in some way.
In the UK (and more widely in the EU) the scope of patent protection available for software generally is very limited. For a good summary of the position in the UK, see:
NB the position under US law regarding patent protection for software is different, with more generous protection for software. Business method patents may also be available in the US, although these were curtailed in a 2014 US Supreme Court case.