In my experience, 「どうも~みたいだ」 has a sense of “apparently”.
In other words, it seems the kouhai likes the senpai, but it’s not definite (to the reader?), so rather than saying he likes her, Urushi says “apparently” he likes her.
Looking online (which risks finding false positives), I came across this Japanese posting which says the English word “apparently” is like 「そうらしいね」 and 「どうもそうみたいだね」 in Japanese, but also mentions 「どうも~みたいだ」 as the kind of situation you would use “apparently”.
Regarding 自分, I’ll quote 's sister site:
Quote collapsed into a details tag due to length.
In other words: 「subjectが自分」 can be tricky to know who 自分 refers to, so you just kind of need to feel your way through based on context.
…is what I get out of it.
I’d figure since the conversation is about the main character, that’s (how we can determine) who 自分 is referring to.
When you see a word that normally ends in だ, such as みたい, and instead it ends in なんだ, that means it’s the explanatory の. I’ve posted an explanation and examples, based on what helped me finally properly grasp the explanatory の.
Story comments chapter 59
What if we found out after all this time that Urushi is actually really bad at shogi, but we couldn’t see it because she had no one to play against except a total beginner?