It seems like you have just a genuine misunderstanding due to language and culture. American discourse uses "right wing" and "left wing" frequently and broadly,

It seems like you have just a genuine misunderstanding due to language and culture. American discourse uses "right wing" and "left wing" frequently and broadly, someone saying they're "right wing" does not mean they are "far right" or extreme or anything like that. Particularly in the context of the original poster, it's a common idiom to state something like "I'm generally left-wing, but I think <insert opposing view on some issue>" as a very clearly centrist person.