The licence defines that you need more than a one site licence if your are redirecting subdomains to another starting root site. As your subdomain is just an API project and not service the subdomain to a different start page then as far as I understand it you can use the same clientsite.com licence for both the web project and the api project. But usually it's best to talk to your EPiServer partner manager for areas around licencing and if you don't have one contact sales.
For a client we're planning to separate WebApi from the website, having two project for these (ex. ClientSite.Web and ClientSite.WebApi) and two urls (ex. http://www.clientsite.com and http://api.clientsite.com)
This requires us to install the packages for EPiServer.Framework and EPiServer.CMS.Core. However our proof of concept results in a license warning being display for api.clientsite.com saying we're missing license.config.
Do we really need an extra site license to run a setup like this ?
The background for separating the API from the website is to be able to deploy the API independently of the website, and also reduce load on the website instance.