I need to add a DBMS to my stack, but I don't know which. I'm tempted to learn SQLite since it would be useful to me with its focus on local access without concurrency. However, doing so feels like I would be defeating the purpose of trying to expand my skill set since it seems like most enterprise applications have the opposite requirements.

To be able to apply what I learn to more projects, what should I try to learn? MySQL? PostgreSQL? Something else? Is there a comfortable middle ground between high applicability and ease of use?

A question you might want to think about is "What kind of experience do I want to gain, by using a DBMS?". If your aim is to have experience with SQL and any related libraries and frameworks for your language of choice (python, I think?), then it kind of doesn't matter too much which you pick so much. As others have said, SQLite would offer you the ability to very easily get started, and would give you a reasonably standard (if a little basic) SQL dialect to work with.

If your aim is actually to have a bit of "operational" experience, in terms of things like what command line tools might be available as standard for the DBMS, understanding how the DBMS handles multiple databases, when to use multiple schemas vs multiple databases, some basic privilege management etc. Then I would recommend PostgreSQL. SQLite's simplicity actually avoids most of these experiences, which is not helpful to you if that is what you hope to learn. MySQL has a few "quirks" to how it manages things like multiple databases, which may lead you to making less good decisions if you tried to take your experience over to different DBMS, especially in bigger enterprise roles. PostgreSQL is kind of a happy middle ground here, with the ability to start PostgreSQL servers via docker or docker-compose making the actual day-to-day management pretty easy, while still giving you experience of the kinds of considerations I have listed above.

At Vital Beats we make use of PostgreSQL, largely because it offers us a happy balance between good management and backup of data, and good standard command line tools, which is essential for us where we are deploying our solutions within Kubernetes / docker, and so more graphical tools are not always appropriate for us. PostgreSQL is also pretty universally supported in terms of language libraries and frameworks, without having to make compromises on how we want to store and layout our data.

For such a more realtime-focused, data-centered application like an exchange, it's not the frontend or backend that matter much. In fact for that, they can do away with any of the popular frameworks like React/Vue/Angular for the frontend and Go/Python for the backend. For example uniswap's frontend (although much simpler than binance) is built in React. The main interesting part here would be how they are able to handle updating data so quickly. In my opinion, they might be heavily reliant on realtime processing systems like Kafka+Kafka Streams, Apache Flink or Apache Spark Stream or similar. For more processing heavy but not so real-time processing, they might be relying on OLAP and/or warehousing tools like Cassandra/Redshift. They could have also optimized few high frequent queries using NoSQL stores like mongodb (for persistance) and in-memory cache like Redis (for further perfomance boost to get millisecond latencies).