He appears to be based on Codsworth, the robotic butler from the video game Fallout 4. Apart from the similar name and British accent, his design is reminiscent of the iconic Protectron robot from the game series or has a shape also inspired by Robbie the Robot.
Conroys have many purposes. For one this preform menial tasks for the family the work for. These can include, cleaning, serving food (or in its case pills), playing games and more. It is also used by the federation to spy on its citizens. When in presence of iLlEgAl tEcHnOloGy it will go into attack mode, where its blue becomes red, and it tries to attack the person.