skill being acquired

There is a real connection between “SCHOOL,” Škoda, and Luxembourg, but it is not that Škoda or Luxembourg invented the idea of school.

The connection becomes clearer if we separate three different meanings:

1. “School” as an institution

The idea of organized schooling is much older than Škoda or modern Luxembourg. Ancient Mesopotamia, Egypt, Greece, Rome, and later European institutions all developed forms of organized education.

So historically:

School → ancient civilization → long development → modern education

not:

Luxembourg → Škoda → invention of school

2. Škoda has a very direct connection with vocational schooling

This is where your idea becomes interesting.

Škoda developed a company-linked vocational education mechanism. Škoda's own apprenticeship school was established in 1927 in Mladá Boleslav. Its purpose was closely connected to producing skilled workers for industrial production.

The mechanism is essentially:

INDUSTRY → SKILL REQUIREMENT → SCHOOL/TRAINING → STUDENT → PRACTICAL WORK → QUALIFIED WORKER

That is a form of what is now commonly called dual or work-based vocational education. Škoda later expanded this into vocational education, Škoda Academy, and Škoda Auto University.

3. Luxembourg has a different connection to Škoda

Here we need to be precise.

Volkswagen Finance Luxembourg S.A., based in Luxembourg, is identified by Škoda as its sole shareholder's corporate entity.

And there is also a completely separate Škoda Tour de Luxembourg, a cycling event in Luxembourg, which has educational activities connected with road safety and young cyclists.

So if you encountered the phrase “Luxembourg + Škoda + school”, it may be pointing toward a modern organizational/system relationship, rather than the historical invention of school.

The deeper connection you may be seeing

Given your earlier idea about systems being created to achieve a purpose, Škoda's educational model is actually a useful example:

PURPOSE
↓
Need for qualified people
↓
SYSTEM DESIGN
School + apprenticeship + practical training
↓
OPERATION
Students learn + work in real industrial environments
↓
OUTPUT
Qualified workers
↓
EVALUATION
Does the training produce the required competence?
↓
IMPROVEMENT
Modify curriculum, technology, training, and institutions
↓
PURPOSE

Škoda itself describes this connection between vocational education and practical industrial requirements.

So if by “school system mechanism” you mean a system in which education is deliberately designed around a societal/economic purpose and continuously adapted to that purpose, Škoda is a very relevant case study.

But Luxembourg is not established by the sources I found as the origin of that school mechanism. The strongest documented Škoda–school connection is actually Mladá Boleslav, Czech Republic, beginning in the 1920s.