In my experience, checkout process UI design is very complicated once you get to the details.
And login/ signup is tricky from legal and technical perspective.
Based on personal experience with checkout design (I have no knowledge of Amazon's reasoning behind their design), considerations that may have played a role:
Many aspects of the checkout process (may) depend on customer details, so if these are not known, it is difficult - if not impossible - to present a clean checkout page.
- For example differences between registered customers versus guests in shipping costs, known preferred invoice and shipping address, known payment details, product price differentiation, payment methods, product availability per geography, free shipping, loyalty, age consent etc etc.
- Checkout design and flow should be extremely distraction free. So if e.g. any shipping address is known, the option/ button to choose alternative shipping address should be minimal, if present at all. This is very, very different from guest checkout, where the shipping address is mandatory, blocking processing until completed. The same goes for payment details, etc etc.That is why it may make sense to start with login/ signup, to keep the rest of the flow distraction free.
One simple unified login/ signup page is easier to maintain than different versions for different purposes.
The entire site may be geared to make users sign-in/ login before checkout. So if site stats show that 90% of users going to checkout are already logged in, then the concession in signup/login is not so costly.
I would consider to maybe add a title to the login page, like "Please login or signup to continue checkout".
But would strongly advise against any other items with links (such as a basket overview). Any additional link/button in checkout hurts conversion, so use them sparingly. Yes, you need to show the customer what they are buying during checkout, but if you show it twice, it will only hurt conversion. So consider carefully where to show it. (login/ signup is probably not the best place).