Connect On-premises⇔AWS in the same segment with Direct Connect (DX)

AWS doesn't natively support stretched subnets between on premises and VPC. There are some partner solutions in this space that will allow you to do that in limited circumstances. Note that this will introduce extra cost to you and add complexity to the network.

But: I would question why you want to do this. Generally, stretched layer 2 subnets aren't a great idea - it's why routers were invented in the first place. What's the use case?

Generally, I see customers ask this question because "I want to migrate into AWS but I don't want to change IP addresses because..." and there are a bunch of reasons that come down to hard-coded IP addresses in applications. In these cases I encourage customers to do the hard yards and fix the root cause of the problem rather than resorting to brittle and (probably) expensive network-based solutions which are just being used as a band-aid.