Yes. You have no more legal right to it than anyone else, unless it is registered in some way.

And if you register things in the name of a company, you had better be sure you will never sell or lose it.

In the UK, unless you keep up the annual registration and pay the fee (and also file annual accounts even if not trading - yet more costs), the company can be struck off, and I BELIEVE that someone else could then reuse the same name. (Would need to check this with Companies House).