Airbus owns the Type Certificate (TC) for Concorde as well as the brand name (thanks to Giacomo Catenazzi for the link). I'm not a lawyer, so I cannot comment on who else is allowed to use the name Concorde (feel free to ask that on Law.SE).
The TC was originally owned by Aérospatiale (created by the merger of Sud Aviation and others in 1970) and British Aircraft Corporation (BAC), but was then transferred to Airbus:
The legal term for the firm is now the “type certificate holder”, in the case of Concorde this was jointly held at first by BAC and Aerospatiale, the two companies that built the airframe. They later past the TC on to Airbus SAS, of which both companies held shares.
(heritageconcorde.com - Concorde Type Certificate and Airworthiness)
At this time, Airbus was not the same company we know today. It was a consortium created in 1970 by several European aircraft companies which continued to exist as separate entities:
The current company is the product of consolidation in the European aerospace industry, tracing back to the formation of the Airbus Industrie GIE consortium in 1970. [...] In 2001, Airbus Industrie GIE was reorganised as Airbus SAS, a simplified joint-stock company.
(Wikipedia - Airbus)
Initially, only Aérospatiale formed part of the Airbus consortium. However, British Aerospace (created by the merger of BAC and others in 1977) joined in 1979:
In 1979, BAe officially joined the multinational aircraft manufacturer Airbus and acquired a 20% share in the venture, the move effectively reversed a decision made ten years prior in which the UK government had withdrawn its support for the Airbus consortium.
(Wikipedia: British Aerospace)
When Airbus was restructured into its modern shape in 2001, parts of the new company BAE systems (the successor to British Aerospace since 1999) were incorporated into Airbus:
BAE Systems inherited British Aerospace’s share of Airbus Industrie, which consisted of two factories at Broughton and Filton. These facilities manufactured wings for the Airbus family of aircraft. In 2001 Airbus was incorporated as Airbus SAS, a joint stock company. In return for a 20% share in the new company BAE Systems transferred ownership of its Airbus plants (known as Airbus UK) to the new company.
(heritageconcorde.com - Who built Concorde?)
Airbus continues to hold the Concorde TC even after its retirement:
Safety rules require every plane to have an air worthiness certificate to keep flying and Concorde’s was held by Airbus. It monitored any safety modifications to the planes and provided technical advice to BA engineers. Airbus thought Concorde should be grounded for good at the end of October 2003 and was unwilling to pass on the air worthiness certificate to another company.
(heritageconcorde.com - Concorde Retirement 2003)
BAE systems then sold its shares of Airbus to EADS (the Airbus parent company, now called Airbus Group) in 2006:
In 2006, EADS acquired BAE Systems' remaining 20% of Airbus.
(Wikipedia - Airbus)
Even if Airbus does not have British partial ownership any more, it continues to operate facilities in the UK, so the British heritage of Concorde isn't quite lost if you ask me :)