one gotcha: schemas are a weird old thing in SQL that are kinda the worst of all worlds, basically more like prefixing your SQL object names than a real level of indirection. Schema objects can't be manipulated as a batch, they don't provide the isolation / access control of databases, and can't be manipulated in DML and require stored procedures to manipulate.
A big advantage of schemas over databases is that you can access different schemas over the same connection, which allows much higher density, since connections are a scarce resource in PostgreSQL. You can also have shared tables.
A big advantage over row-based sharding is that you can have variations of the table definitions or a different set of tables altogether in each schema. Plus for the purpose of sharding it's easier to use, since normalized data models complicate row-based sharding.
I would make a partitioned table, and put each partition in its own schema.