Personality trait in which the person's style is evident regardless of the situation? What is the term for the personality trait in which a person's style is evident in all they do? The person's particular behavior patterns expresses the attitudes and inner characteristics of the person, regardless of the situation. A person with this trait puts a little bit (or a lot) of themselves in all they do. In contrast, other people may blend in more and adapt their behavior more strongly to what is expected by others.

I once read about this term in a personality model article online and have never seen it since.
You may be referring to people who are low in self-monitoring.
Wikipedia has a good summary:

Self-monitoring is defined as a personality trait that refers to an ability to regulate behavior to accommodate social situations. People who closely monitor themselves are categorized as high self-monitors and often behave in a manner that is highly responsive to social cues and their situational context. High self-monitors can be thought of as social pragmatists who project images in an attempt to impress others and receive positive feedback. Conversely, low self-monitors do not participate, to the same degree, in expressive control and do not share similar concern for situational appropriateness. Low self-monitors tend to exhibit expressive controls congruent with their own internal states; i.e. beliefs, attitudes, and dispositions regardless of social circumstance. Low self-monitors are often less observant of social context and consider expressing a self-presentation dissimilar from their internal states as a falsehood and undesirable. People who are unwilling to self-monitor and adjust their behavior accordingly are often aggressive, uncompromising, and insistent with others. This may make them more prone to condemnation, rejection, and the possible consequent feelings of anger, anxiety, guilt, low self-concept, isolation, and depression. Even the occasional indiscretion can make social situations very awkward, and could result in the loss of a friend, co-worker, client, or even job. Those who are willing to adjust their behavior will often find that others are more receptive, pleasant, and benevolent towards them.

Self-monitoring is usually measured with Snyder's (1974) Self-Monitoring Scale or variants thereof.
The scale measures agreement to items such as:

I find it hard to imitate the behavior of other people.
My behavior is usually an expression of my true inner feelings, attitudes, and beliefs.
At parties and social gatherings, I do not attempt to do or say things that others will like.
I can only argue for ideas which I already believe.

References
Snyder, M. (1974). Self-monitoring of expressive behavior. Journal of personality and social psychology, 30(4), 526.