Different kinds of language proficiency

I'm also trilingual: I speak Russian, Hebrew and English, in this order of when I learnt them. I feel that my proficiency in neither of them is Pareto dominated.

My pronunciation is worse in English than in either Russian or Hebrew, because I learnt English mostly from reading and writing rather than speaking and listening. My spelling is worse in Russian than in either Hebrew or English (despite Russian spelling being objectively the easiest), because, while I spoke, listened and read a lot, I rarely had to write anything. My vocabulary is worse in Hebrew than in either Russian or English, because I read a lot of books in Russian and English but only a few in Hebrew. And also, like Self-Embedded Agent said [LW(p) · GW(p)], the vocabularies are not strictly comparable.

However, I don't think that for me these languages differ a lot in terms of emotional closeness. Maybe it's because I never felt like I fully belong in any particular ethnic culture.

Another thing to note that your Swedish & Finnish and English vocabularies might not have a proper inclusion relation. My technical & academic vocabulary is larger in English than in my native language, but for household appliances/vegetables/flowers it's the other way around. 

I speak Russian, Ukrainian and English and am slowly learning German. I also have studied and forgot Tatar, Hindi and French (there was some rather twisted education planning involved.)

I feel like the languages occupy their own niches, even the forgotten ones. German feels like really elaborate English with the promise of expressing thoughts more nimbly and precisely. Tatar helped Hindi through some words of the same "Turkish" origin (I mean words like "kitap(b) - a book"), and Hindi was fascinating in its unlikeness to most anything. I still have this "standard of otherness" in my mind. French... is like knitting phonemes, it was taught to us a process more than a language. English is for building from sometimes not-intuitive blocks, and yes, there is that chasm. Russian is my tool for "fencing", "being witty", easy to use like a well-balanced knife. And Ukrainian is what I do :)

We were told, when we began studying English, that the more languages you know the more times you're human. Still, nobody said we're gonna like the end result, for some reason.

I have similar experiences as those described in the post and in the other comments.

I speak Dutch, English and German, all of them to an advanced level and all of them currently in use. I learned each language in a different way and use each language in a different situation. But they are closely related languages and the differences are subtile. My vocabularies in each language do not fully overlap, and knowing three languages gives me access to a wider range of words and phrases - which I unfortunately cannot share with most conversation partners.

Dutch is my native language but since I left the Netherlands I don’t use it in everyday work or life but do use it in personal notes and calls with my family. I can flow into a Dutch conversation with ease even if I haven’t been exposed to the language for weeks. Similar to Kaj, I do feel this emotional connection with words and phrases in Dutch that is lacking in any other language. Dutch phrases can come across as unusually friendly in some subtile way. I like to talk in Dutch whenever there is an opportunity, although not all native Dutch conversation partners I meet seem to share this.
Reading and writing in Dutch - somehow - feels awkward and unnatural. Discussing things like rationality, effective altruism, and software engineering in Dutch even more so. I spend a lot of effort finding my words, and once I got them, I want to throw them away because nothing quite fits.

My experiences with English and German are also different from each other. I learned the basics of both at school, but became advanced in English by reading university textbooks and I became advanced in German by socializing with native speakers. There are many everyday objects - household appliances, bike parts, tools, foods - that I know the name of in German but not in English. I have trouble understanding conversational English from native speakers.

I live in a multilingual area and it’s common that when you meet a person for the first time, you agree on which language to speak and stick with that, also if you have more than one language in common. And if there wasn’t a norm that conversations should be held in one language, I would mix much more so we could access our larger shared vocabulary.

English natively, then learned Spanish in school, all throughout primary, secondary, and university.  I often dreamed in Spanish and could communicate fluently.

The emotional valence you're talking about is something I never experienced, probably because I never trained it.  All of my Spanish was academic, even the literature classes I was taking in college.  I had very few organic Spanish conversations with Spanish speakers, and I never needed (or had the opportunity) to use Spanish to build a friendship or court anyone.  

The few times I was purely socializing with Spanish speakers for (say) an hour or more, the Spanish part of my brain kicked into overdrive, and I was even more fluent than normal--like the English part of my brain was just gone, and I was meta-aware of that feeling, but meta-aware in Spanish.  It was similar to a flow state.  The chasm was definitely gone in those situations.  Or at least it was for me--maybe my Spanish was just bad enough for the other person that I was in an uncanny valley (uncanny chasm??).

It's interesting to me that you feel a stronger emotional connection to a language you speak less well – and other commenters seem to have a similar sentiment. I think I only have that for very emotionally charged statements (i.e. "I love you") and not for normal conversation. I usually prefer to talk in the language which I have recently used least, and also the language where there's most native speakers present. When I'm confused about the language spoken, I default to English – but I still get a weird feeling if I'm speaking English to a group which consists exclusively of other German native speakers.

Maybe the language we grew up using is rooted deeper in our psychology.  Children's stories, fables, songs, & the music of our native language connect our young brains to the world and, when we remember those things, we get that feeling we might call 'nostalgia.' Nostalgia is something, I think, that only comes after many years... Maybe one day when you are older you might hear English words, stories, music that give you this feeling.