Technically both these projects are infringing trademarks. SQLite is a trademarked term by the SQLite authors. OP is not more wrong than https://sqlitestudio.pl/. If https://sqlitestudio.pl/ is using the mark under license, they haven't indicated that anywhere (usually a requirement).
The SQLite author says in the last paragraph of the sqlite-kit issue:
"You are welcomed to use "SQLite" as part of the name for your software, so that people know that it uses SQLite. For example, you might call your software "SQLite Wrapper for Swift" or "Vapor's Swift SQLite Wrapper". But you may not use the unqualified name "SQLite"."
There is on the one hand what the Author wanted in this case, and what the law is on the other. He may have many reasons for not enforcing his rights to the maximum extent.
He may want to be nice, or have done a deal or any number of reasons. That doesn't mean he can't stop them from using it altogether.
Also he can change his mind down the track.
Whether it is infringing or not is up to many factors in each case, but simply extending it is not going to protect you in every case.
If we want a definitive answer, would have to take this to a court.
Source: studied IP in Law school, but not a professor or anything.
If this were not the case and a trademark is USA specific, how is is to be assessed if, as in the case of SQLite Database Explorer, someone from Ethiopia or, as in the case of SQLiteStudio, someone from Poland violates it? Or the other way round, if someone from the USA violates an Ethiopian or Polish trademark?
I went to law school in Australia over 10 years ago so things maybe have changed.
At the time we were taught about the Madrid Protocol (US implementation https://www.uspto.gov/ip-policy/international-protection/mad...), which gives you some time to register it in each country after the first registration. If you register it within this alloted time, you retain priority over other people who want to register it too.
You do have to register it, but more importantly, enforce it, yourself worldwide.
IP Lawfirms usually provide this as a service.
On the practical matter of "Is the trademark registered internationally?". Have to do a search in each country's trade mark office. It's very tedious (not every mark is in text, some are pictures, sounds).
[0] https://sqlitestudio.pl/.