Actually, Boeing design and assemble Boeings, and Airbus design and assemble Airbus.
Most of the parts, even major sub-assemblies are manufactured by 10001 companies worldwide acting as skilled sub-contractors. For example, some years ago, I was working for Westlands Helicopters - and spent most o my time on a major sub-assembly for the Boeing 777 !
That said you should see the monstrous Airbus wing factory at Hawarden, or their assembly plant in Toulouse - astounding bits of Engineering in themselves.
G