I consider myself functional, but not proficient in SQL. Is there a reliable and easy way to assess one's skills in SQL? I feel that most of the new things I learn about SQL these days are database specific. I wonder whether I'm missing out on something, or do I already have the "core" SQL knowledge down, and everything else is special cases?
The core concepts are the most important things to know. How do SQL queries and their various clauses work. Tables, data types, indices, and constraints are all very important. With that knowledge you can handle most SQL databases, and begin to learn and work around their quirks. Yes, there are a lot of database specific details. Procedure languages can be useful such as PL/pgSQL for PostgreSQL and T-SQL for SQL Server.
I recommend you look at one of Markus Winand's presentations, such as this one : https://youtu.be/xEDZQtAHpX8
I felt it summarized well the modern foundation of SQL databases.