Word for a person with the same opinion on everything?

Solution 1:

There is a somewhat broad phrase, Johnny one-note (or one-note Johnny), that covers repetitive views beyond simple affect

An opinion or viewpoint on a single subject or a few particular subjects which is frequently expressed by a person or organization

It derives from the song, "Johnny One Note", in a Broadway musical

a show tune from the 1937 Rodgers and Hart musical Babes in Arms. The lyrics tell a story of a male vocalist named Johnny who sang with a band and could sing only one note, but sang that note extremely well.

The person you describe is an emotional Johnny one-note.

It's a bit less colorful, but you could simply use the term one-note

unvarying in tone or emphasis : monotonous

Solution 2:

I was going to post it as a comment, but looking at the definition in Urban Dictionary, it could be closer to what you want then I thought.

One Trick Pony - used to describe a friend, acquaintance or stranger, who has very few talents, sayings, jokes or skills. _{Definition on Urban Dictionary}