Hi,
YC resolves shop by domain name, because paths are unreliable given the complexity of rewrite rules that are possible. Domain defines the shop and this is the only way it can work.
You referred to our demo urls:
but these are NOT different shops, these are 2 web applications, the regular YC and enterprise edition, but the shop is the same demo shop. For example try adding some items in your basket and then change URL, you will see that in the other application the basket will have same products. This is because both applications serve SINGLE shop. So it is not 2 shops but only 1. However the applications are different so you get different features on each one.
To have completely different shops: i.e. separate catalog branch, separate design, separate products, separate customer account etc, you MUST have different domains.
This is how YC works.
Now if you really want to have it your way. The only thing you can do is have 2 different war files. And each file has to connect to separate database. So you will need to databases as well. And then deploy both of your war files under same tomcat. The you could have what you want BUT it will be 2 completely separate databases. So you will not be able to share data between shops. In terms of YC those would be 2 completely separate instances.
Regards,
YC team