teaching and testing MySQL's relationship with the ISO SQL standard has always been, err..., problematic. I will give them credit for having become better in recent years.
portability, you can use SQLite - it's a rock solid, very standards compliant database which will run off a memory stick. There are a number of videos from the creator of SQLite on YouTube!
SQLite is a C-language library that implements a small, fast,
self-contained, high-reliability, full-featured, SQL database engine.
SQLite is the most used database engine in the world. SQLite is built
into all mobile phones and most computers and comes bundled inside
countless other applications that people use every day. More
The SQLite file format is stable, cross-platform, and backwards
compatible and the developers pledge to keep it that way through at
least the year 2050. SQLite database files are commonly used as
containers to transfer rich content between systems 1 3 and as
a long-term archival format for data . There are over 1 trillion
(1e12) SQLite databases in active use .
From here, SQLite is the "Protégé of PostgreSQL" - its primary developer, D. Richard Hipp is always up front about how he
models copies PostgreSQL syntax.
It's approx. 2MB (yes, MB, not GB!) and is probably better (for teaching purposes), and certainly more standards compliant, than MySQL. See this page for a comprehensive (and very honest) assessment of the pros and cons of SQLite - in your use case, I would pay particular attention to the
Education and Training section. It is gratifying, in these days of endless marketing jargo-babble to see a vendor1 be so up-front and honest about a product!
1SQLite is not a vendor in the classic sense, it's public domain software, however if you watch D. Richard Hipp's videos on YouTube, it rapidly becomes clear that he makes his crust from servicing the needs of commercial clients, of which there are many!