For example can I start producing exclusively designed plungers and name them "Colgate" (like the toothpaste), write my own Internet browser (to compete with Google Chrome and Mozilla Firefox) and name it "Orbit" (like the chewing gum) or write a book, say, about how to organize your life piece by piece and name it "Lego"?
There's a famous case that discusses this very thing. Back in the ancient days before millennials roamed the planet, there was a record company, Apple Records, it was the publishing/record company of The Beatles. Along came these two fresh young kids from Cupertino, who had just started Apple Computers. Apple records sued for trade name infringement.
Ultimately, they settled, recognizing that they were in different industries and the chance of dilution was minimal, and Apple computers agreed to stay away from the music business.
Ten years later, the new Macintosh computer had a sound card and could play music. Apocryphally, Woz has been dinged for naming the default Macintosh alert sound "SoSuMe" ostensibly intended as a raspberry taunt to the moribund Apple records...
Maybe; or maybe not. It's complicated.
There, they point out:
Use for different goods or services
The registration of a trademark includes an indication of the goods or services which it is intended to protect. This means that, in principle, others are free to use the trademark for other goods or services. However, there are some exceptions. As explained earlier, a trademark always runs the risk that it loses its distinctive character, which could mean that the trademark at some point is annulled.
It is also an infringement if the use of the mark is such that it harms the trademark holder in an unfair way. The reputation or image that he has built could suffer from somebody elses use of the mark. For example, the Dutch holder of the trademark King (who makes peppermints) was able to successfully stop someone else from selling condoms under the same trademark.
So my (non lawyer, lay) read of this article suggests that, generally speaking, use is permitted for different goods or services unless the situation is subject to exception. Words describing the exceptions like harm and unfair make the answer subjective and situation-dependent.