It has nothing to do with the underlying OS. It has to do with the collation of the instance and the collation of the database. Here is an example that shows that database names, table names, column names, and data are not subject to case sensitivity by default.
If you want those things, you can turn them on, by specifying a case-sensitive collation for the column, for the database, or - I think - for the instance. (The images I've been using for SQL Server Docker containers don't have the traditional setup package where you can pick instance collation prior to setup, and changing instance collation after the fact might not be fun.)
Perhaps you're thinking about physical files? I know when I was moving my stuff from IIS to a Unix host I had to change references to things like
file.css because the file on disk was actually
File.css and the server wouldn't be able to locate it. SQL Server doesn't work that way.
Yes, you should absolutely stick to the proper casing of things, and that future-proofs you no matter what operating system you use to host SQL Server. Anybody can come along and change the collation of various bits of your installation (or deploy your code to a place where the collation is already different). See these articles: