Case sensitivity/insensitivity is a property of the collation, which is configurable on a per-column basis. In Postgres the default collation is case-sensitive

Case sensitivity/insensitivity is a property of the collation, which is configurable on a per-column basis. In Postgres the default collation is case-sensitive, and in MySQL the default collation is case-insensitive, but this does not mean one is "correct behavior" and the other is not. MySQL's out-of-the-box collation support is arguably a lot more thorough than Postgres's!