It is possible, in fact a few seconds searching brought up one example. Disclaimer: I have not used that plugin.
The first thing to ask is: what is Eclipse, exactly? What does it provide?
Eclipse is an IDE. It provides frameworks which can be leveraged by plugins which run in the IDE. It is best-known for being a Java IDE, and the Java plugin is one of the oldest and most mature available for Eclipse. But you can do anything in Eclipse: you could even implement a database, photo editor, or video game if you wanted to. In fact there is an entire RCP framework for building arbitrary (non-development, even) applications.
What does the Eclipse Java plugin (JDT) do?
JDT includes two components relevant to this question: a Java editor and a Java compiler. The two work together to provide those red squigglies (their real name is "error markers" but I prefer squigglies). In the background, as you edit Java source code, the JDT compiler is building an AST (abstract syntax tree) that is basically a data structure that represents the partially-compiled code in memory. Its purpose, in this context, is to identify regions of your code: variables, method names, class declarations, etc. This is how JDT can tell you "missing
}": the AST is broken, and it expected a
} but you did not provide one. This is also how if you select a method reference you can hit F3 to go to its definition.
How would a LaTeX editor in Eclipse work?
Now we get to the core of your question. Clearly, a LaTeX editor would not work the same as a Java editor because LaTeX has no "compiler." Java and LaTeX are just different. But is there any way to correlate them?
You would need something similar to the background compiler in JDT, but not a compiler. Some process that runs as you edit your document, checking for errors. This is more difficult as David Carlisle mentions because LaTeX has a more complex grammar. Java (and most programming languages) was built to have a grammar that is easy to parse, making the compiler easier to write with the side effect of making the language more programmer-friendly.
LaTeX is a lot older, it has evolved differently from programming languages, and its grammar was not designed the same way a programming language grammar would be. But clearly it can be parsed, because we have programs that consume LaTeX.
Specially, in the second question, in the selected answer, it has been
claimed that writing such an editor means a background-running editor!
If it is surprisingly the case, what is the intrinsic difference
between Java and TeX which makes this difference?
I hope my explanation above demonstrates that there is no difference. JDT has a "background editor" if you will (actually an editor and a background compiler working together). A "smart" LaTeX editor would need an editor... and a background syntax checker. But in the context of an editor, the JDT compiler is basically performing the exact same role.
!on the suggestion of running tex in the background, any implementation of what you ask will surely do that. Also I don't understand your reference to Java, surely you don't run your java program as you type it that sounds very dangerous.