A mismatch in formality is noticeable too in English, from my experience. It think it's a bit more subtle compared to other languages, as there is no formal "you" (in contrast to "du" versus Sie" in German, "tu" and "vous" in French, "tú" and "usted" in Spanish, "你" and "您" in Mandarin, and examples in other languages).
However, one difference in English is with greetings. If you are coworkers, especially at around the same age and seniority, it feels unnatural to say "Dear ___," versus "Hey ____" or "Hi ____" (experiences may vary based on organizational culture). It's noticeable with texting, too. If you were to use formal punctation and end all texts with a period (such as: "Hi." or "I'm fine."), the texts may be perceived as less friendly (in contrast to: "hi" or "no prob, it's all good").
I believe that informality in English communication is often a signal of how close the relationship is. Though some people speak with others with a high level of formality even in their close relationships (or alternatively, speak informally in nearly all contexts), a change over time from formality to informality is often a sign of growing closeness and trust. Long story short, this plays out the same as you've described the effect in English, at least for personal relationships.
For professional relationships, it depends on the person and the organization. I generally try to roughly mirror the level of formality of the person who I'm speaking with. However, if I'm reporting to a person with higher seniority than me, I do try to have at least a minimum level of formality. Even if a CEO were to send a text in a very friendly and informal way, I would try to be polite but at least semi-formal.
More explicitly, the Korean language makes the different possible levels of speech clearer. This source [1] identifies five to six levels of formality in Korean (Wikipedia identifies a higher seventh level reserved for historical dramas, addressing royalty, and religious texts). So, even if a coworker were to address me with the Korean equivalent of "plain" or "intimate" speech ("heya {name}", all in lower case), I've found it the most comfortable to respond in at least a familiar or polite way ("hey {Name}..." or "Hey {Name}"...).
That way, there is both a sense of respect and a sense that there is an attempt to match the person's level of familiarity. There is then less of a chance that a person might feel uneasy due to either too much familiarity or excessive formality on my part, which makes communications easier.
[1] https://socialsci.libretexts.org/Bookshelves/Anthropology/Li...