Are some spare parts made by Airbus?
Why are they taking the wing to Toulouse, FR?
Because the part was found on the island of Réunion, which is French territory.
Why is a part from a B777 being analysed by Airbus?
It is not. Airbus is not the only company/institution to have plants/facilities in Toulouse: The BEA (France's equivalent to the NTSB) also has facilities there.
To extend a bit on Federico's answer. As I understand, BEA, the French accident authority, have some office in Toulouse while the facility itself where it will be looked at is military.
France's BEA crash investigation agency will examine the wing flap at the Direction Generale de l'Armement compound near Toulouse. Source
Also from the BEA website, under the heading ICAO:
In particular, States must start an investigation in case of a civil aviation accident or serious incident on their territory. It is specified that "The sole objective of the investigation of an accident or incident shall be the prevention of accidents and incidents" and that "It is not the purpose of this activity to apportion blame or liability".
Close association between the representatives of the States of Registry, Operation, Design and Manufacture of the aircraft is organised. This brings to the State that is leading the investigation useful skills and information and allows for feedback that is as fast and complete as possible in terms of safety. Source
Which explains their responsibility in investigating the part. I don't think there are too many facilities with extensive knowledge of forensic engineering on impact-damaged (composite) parts (Even analysis of damaged Black Boxes is often 'outsourced') so they might even be happy to have the large BEA look at it.
The French are not unfamiliar with investigating aircraft accidents and have participated in a few substantial ones in the last few years. I discovered presumably the same folks were involved in forensic engineering work of the AF447 remains in 2011, which also crashed into the ocean.
French investigation agency Bureau d'Enquetes et d'Analyses said the vessel would transfer to Bayonne, a port on the south-western coast, and parts from flight AF447 would be moved to a hangar in Toulouse.
The wreckage would be examined by the French defence ministry's technical aeronautical division. The recovery effort also retrieved several victims of the crash and these will be taken for forensic assessment.
Footnote: Since accidents with unclear events (or presenting controversial claims like crashing the plane) such as this one which can be a hot potato it is sometimes even desired to have a independent third party look at the stuff.