> catalog, for example, is pretty meaningless on H2

Yes. It's just a compatibility feature.

> is "auto_increment" the same as IDENTITY?

Both identity and auto-increment columns are columns with a sequence as the default. The column declared as the identity columns is implicitly the primary key column of this table (unlike auto-increment columns).

> Also, for curiosity, given create table test(id int auto_increment) what is the name of the generated sequence? Is there a pattern?

The name of the sequence is "SYSTEM_SEQUENCE_" and then a globally unique random number (a UUID). The reason is to avoid collisions, which could occur if the sequence name is generated from the table name, and then the table is renamed. Sequences are not renamed if the table is renamed.