Yes, if you have DNS service at Domainnameshop (ie use our name servers), then ANAME records are supported.
According to the DNS standard, it is not permitted to have a CNAME record directly on the root domain, i.e. you can have a CNAME for www.mittnavn.no but not for mittnavn.no. At Domainnameshop, we therefore offer a variant of CNAME called ANAME, which in practice provides the same functionality. With some other providers, this is called ALIAS or CNAME flattening.
If you try to create a new DNS record on the root domain in our control panel that points to a host name instead of an IP address, we will automatically create an ANAME record. (If you use "Advanced settings" for DNS, you must choose ANAME instead of CNAME in the drop-down menu, otherwise you will get an error message.)