Is it possible for me to have an alias (in Route53, for example) to the default environment? I tried adding the CNAME record as indicated, but because of the domain being the apex, I am not allowed to add a CNAME record.
Is there anything else I can do or do we just get the default name?
Not for the default environment, because it doesn’t support custom domains. All other environments do. This may change in the future. In any case, the default environment always points to the latest deployment on any branch, so it changes frequently and for that reason you probably want to avoid using it for established projects. It primarily exists to help free users get up and running quickly.
Regarding Apex domains in general, Route 53 requires you to create A records instead of a CNAME. See: