The consistency guarentee assuming repeatable read or higher on each shard is parallel snapshot isolation, which is technically still higher than postgres read committed.
If you have serializable on each shard and you chop transactions so you have no cross shard transactions, the application is technically still serializable, it just behaves like two independent deployments of the application for different shards
If you have serializable on each shard and you chop transactions so you have no cross shard transactions, the application is technically still serializable, it just behaves like two independent deployments of the application for different shards