Both of my parents and I are multilingual. We come from Czech Republic and have learned multiple languages through our lives as we moved around.
I have noticed, even within myself that I "feel" different speaking a European language compared to English.
Not being able to go into the neurobiology of brain plasticity and change within when developing with multiple languages I can give my understanding through Psychology.
I think that language can be contextual in different parts of a persons life if it is a progression in time from one language to another as opposed to learning multiple languages at the same time. This in turn can build associations that bind the language with emotion at the time of learning the language.
For example, when I learned Czech until I was 7, which I have a different emotional connection and association with. So when I hear it and speak it I feel different. I feel calmer and my stress level decrease, I also realised I have different reactions to circumstances.
However when I deal with day to day life in English I feel more assertive and actually aggressive in my speech and reactions.
I believe that the learned associations between the language and the persons life can create a type of personality that isn't necessarily a split personality as it is the same cognitive person.
Sorry if I used the wrong expressions in trying to describe the scenarios from my own experience and introspection on the same topic.