That is true for any domain name you purchase through a host, though. You are the owner - but do they give you control over it?
My business does actually register domain names for our clients. I do it because they ask me to do it and they can’t figure out how to do it themselves. If and when they decide to leave us, I have to take an active part in getting the name transferred to a different registrar because we don’t make a habit of taking care of domain names for people not hosting with us.
My clients don’t have their own domain registrar login and can’t edit the domain info themselves. I just have one account for the company at godaddy.com with about 80 domains.
It is not a huge deal for me to take care of customer transfers - when they transfer away from us. I know how to do it. We’re a small company so I’m easy to reach and easy to work with.
However, getting a domain AWAY from another hosting provider has proven to be an absolute nightmare over and over again. We typically have to get control over the domain to get the nameserver changed to our nameserver so we can set up hosting. Bigger companies either seem to have terrible customer support in this area or like to be “bullies” about it because they’re losing a hosting account. Getting domain names transferred is actually a seriously stressful part of my job.
Before going to 1and1.com and getting a whole “package deal” with hosting and domain names, check with them to see if you have a registrar login and are able to edit your domain information yourself. Alot of companies DO do this, but some don’t, and that’s where the trouble arises.
If they indicate that you can’t edit your own domain info, I would strongly advise that you register the domains yourself at godaddy.com so if you do have to pack up and leave them, it will be much easier.