What controllers are used in Airbus/Boeing fly-by-wire systems?
The systems used by complex aircraft are typically made specifically for an aircraft platform and all the different manufacturers of both air-frames and the control systems will have slightly different system architectures.

For example, a system will have a flight control "computer" made by one manufacturer, this will send commands via a data bus to the actuator control unit which will move the control surface via a hydraulic actuator.
Another may have the flight control computer directly control the actuator.
Remember that with flight control computers there will be multiple computers all performing the same task, providing redundancy should one fail. Duplex and Triplex flight control systems have historically been used.

Companies like BAE Systems, Thales, Rockwell Collins, GE Aviation, Parker Aerospace, Safran are just some examples of the many companies involved in the complex supply chain for aircraft parts.

Flight control systems for a single aircraft may be made by separate manufacturers to ensure that no single design fault causes a catastrophic failure or loss of control. This is known as dissimilarity. One manufacturer might make a flight control computer with a PowerPC processor and another will use something totally different.

In other cases, the same ASIC (Application Specific Integrated Circuit) will be specified to two separate integrated circuit manufacturers who will both build the same part.

This interesting article covers some of the points covered above
https://www.aviationtoday.com/2001/10/01/a380-more-electric-aircraft/