For mathematics, in English, we have "even" numbers and "odd" numbers. Even = 2 4 6 8 etc. Odd = 1 3 5 7 9 etc. All sports have an even number of players. If 22 people want to play soccer, then 11 will play for one team and 11 will play for the other. If 12 people want to play hockey, then the two teams will have 6 players each. If you have 23 people who want to play soccer, then one of them is the "odd one out" because the teams can no longer be "even". This is true for many things in life. The tables in a restaurant will typically have 2 seats, or 4 seats. The number of seats in a theater row will be an even number, perhaps 30 or 40. Most cars are designed to carry 2 or 4 people, not 3 or 5. Being the "odd one out" doesn't mean you are being strange or different from the other people in the group. It just means that you are an "extra" person in a situation which is not designed to accommodate you. So if 2 people are playing tennis and you want to join the game, you will be the "odd one out". If 5 of you are going to dinner but the restaurant only has a table with 4 chairs, one of you will be the "odd one out". If your stepfather wants to be with his wife, but he doesn't want to be with the child, then he will make the child feel like he is the "odd one out".

It means he never belongs, never fits in, is never accepted. This term is often used in quizzes and IQ tests. An obvious example: 1,3,5,B,9 -> Which is the odd one out? (Or which doesn't belong in this group?) Of course, that'd be the letter B, because he test are numbers. In a similar sense the narrator feels he's always left on the sideline, that he's never accepted in the group, family, etc.

sigficase the extraño papi like the odd one out

For example, if a man is in a group of females, the man is the odd one out. It is neither good or bad (in most cases) and simply used to describe something different from the others. I have to admit, the usage of the phrase in your example seems a little weird, maybe I just have not seen it used this way.