is that bias coming from the programmers themselves or is AI itself inherently biased

it comes from the data used to train it. Which is theoretically chosen by the programers but is so long that no human could realistically read through it.

AI doesn't use logic to come to conclusions it uses statistical probability to generate sentences which put the words in the right order to mean something in english (the AI doesn't understand the meaning of anything it says and it is incapable of such understanding) and uses statistics to associate responses as relevant to prompts

Being right wing is not logical at all if anything Socialism is rational as Socialism is the system which has selected an end it considers good and advocates doing the practical things to achieve this. Which is rational thinking. Capitalism on the other hand wants to destroy the planet to make crap we throw in landfills. This is irrelevant however as the AI we are talking about here is not using reason to reach its conclusions