Yes, today anyone can register a domain name that is vailable for purchase and doses not infringe any third party’s rights.
The domain names were initially registered by Network Solutions only, that had the sole monopoly to register the said TLDs (Top Level Domain Name). This monopoly of Network Solutions continued for many years and only in 1999, the Internet Corporation Assigned Names and Numbers (ICANN) allowed other accredited registrars to register domain names. Today, there are more than 100 registrars with whom one can register a TLD.