We already have a .com domain name (or other top level domain) registered. Is it possible to register an identical .si domain name?

You can register any domain name that meets the requirements of The General Terms and Conditions for Domain Name Registration Under .si TLD regardless of what you have registered under other top level domain names. If, for example, you have registered a domain name xyz.com, you can easily register xyz.si domain name as long as it is not yet taken.