I’m not sure if there is some other meaning for the phrase object storage, but my take on it is that object storage is used when referring to an API that can service arbitrary data given a key path. Minio and Riak are selfhosted services capable of providing S3 compatible object storage. The data you put over HTTP is arbitrary, the API doesn’t care if it conforms to any file format. It’s also not a file system.
Block storage is the raw block device that a file system lives on top of. Most CNIs will provide you with a file system that it already allocated for you, sometimes it does so overtop of a block device that it will automatically manage for you. There are a few cloud provider CNIs that wont present raw block storage at all.