Owning a brand name is narrower than it sounds. Protection is granted within defined boundaries, and outside them the same name may belong to somebody else.
Registration is by class and territory
Trademarks are registered against categories of goods and services, and the rights obtained apply only within those categories.
They are also territorial. A registration in one country confers nothing in another, which is why international protection requires separate filings or a regional arrangement.
This is how identical names operate legitimately in unrelated sectors and in different markets. Neither owner is infringing because their protections do not overlap.
Distinctiveness determines what can be protected
A name that merely describes the product is difficult to register, because granting exclusivity over ordinary descriptive words would prevent competitors describing their own goods.
Invented words receive the strongest protection, followed by existing words applied to unrelated products. Descriptive terms sit at the weak end and often cannot be registered at all.
Which is why the easiest names to market are frequently the hardest to own, and the trade-off has to be resolved before the name is used publicly.
Rights depend on use as well as registration
In many jurisdictions a registration can be challenged if the mark has not been genuinely used in commerce for a defined period.
Some systems also recognise unregistered rights arising from established use and reputation, which can defeat a later registration by someone else.
The balance between registration-based and use-based rights differs considerably between legal systems and changes over time, so the position is jurisdiction-specific.
Enforcement is the owner's responsibility
No authority monitors infringement on the owner's behalf. Unchallenged use by others can weaken a mark or, in the extreme, cause it to become generic.
Owners therefore watch registers and marketplaces and object to conflicting applications, which is an ongoing administrative cost rather than a one-off registration fee.
Enforcement is also a commercial judgement. Pursuing every minor use is expensive and can attract more attention to the infringement than ignoring it would.
What a mark does not cover
Trademark protects identifiers of origin: names, logos, sometimes shapes, colours or sounds. It does not protect the product, the idea or the business method.
Those fall under other regimes with different requirements, durations and costs, and a business often needs several to protect what it thinks of as one thing.
Understanding which regime covers which asset determines what a competitor can lawfully copy, which is a strategic question rather than a legal formality.