ASE can have up to 32K dbs (includes various system dbs).
master db holds server wide metadata (eg, logins, roles, configs, databases) and can also hold user data (though this is discouraged).
Other system databases include
sybsystemprocs (collection of system stored procs and functions),
tempdb (temporary scratch space; DBA can create additional temporary dbs),
model (template used when creating a new database),
sybsystemdb (used to manage distributed txns); there are several other system dbs that can exist depending on options/features the DBA enables.
Any other databases would fall under the heading of
And yeah, a user db would be similar to a HANA tenant db, though all ASE databases are accessed via the same host:port combo (HANA tenants have their own port if I recall correctly).