We all speak (and behave) differently when we are with our parents, our colleagues and our friends - and arguably online, even when speaking in our native tongue in all these (and other) scenarios.
That is not to say that certain languages don't lend themselves better to certain forms of expression, but I'm not convinced your anecdotes support that argument.
All my friends were Japanese until middle school, and my social life was dominated by Japanese. Definitely didn't fit into American elementary school at all and was a problem student.
But then I stopped going to Japanese school in HS, and then figured out how to excel in American school, met likeminded friends for the first time, etc.
Then I went to work at a Japanese megacorp in my early 20's.
It's a strange mix of time spent using the language.
In a vacuum, it's a net positive to be able to understand an additional language very well. But I've felt that my life would be easier in Japan if I were "more foreign", since I wouldn't be subjected to the expectations of a domestic person while actually internally being a foreigner.
Also I'm convinced that had I spent the resources that would put towards Japanese into English, I would have become much more proficient at English earlier in my life. That would have translated into better verbal test scores, and may have meant I wouldn't have had such a hard time in my younger years in school. Then again plenty of solo-English speakers of the shy nerd bretheren have problems at school of not fitting in, so this isn't only a language issue.
Honestly it's just a grass is greener type of situation. It's brought me great advantages (I wouldn't be doing my current company without this advantage, and I genuinely think the language is a beautiful and sonorous one, and the cultural traditions are admirable), but it's also held me back in many ways compared to my 100% American English peers as well.
Practically every English speaker I know who speaks Japanese well has a HUGE mode shift when changing into and out of Japanese.
And it's oddly specific to Japanese. It doesn't seem to be a language difficulty issue as this doesn't seem to hold for the people who speak Chinese (Mandarin or Cantonese) or Korean.
Not necessarily parents, colleagues, friends but cultural differences between countries/regions. I'm a Swiss living in Colombia. When I speak Spanish here my talking style differs quite a bit from home, because I'm adapting the local customs. But I'm still me. I can't think of anything I would do or not do, deem good or bad, just because I'm using another language.
Where language comes in is more in personally projection. Because of different forms of expression which are not 1:1 mappings and also because I'm not as eloquent as in my native tongue, you might think I have different personalities.
On the other hands I have several Finnish personality "traits" although I don't speak the language. But my mother is from Finland and I grew up with plenty influence of her culture.
I suspect, that if the language spoken has influence on the personality it is rather small or entirely emergent from the associated culture and time of life we learnt/used it.
It's the old mistake of taking correlation for causation.