Yes, but Elasticsearch requires a dedicated server, you may have to use a river, you may have to shard, etc. Some people (most people?) don't want to think about shards & rivers and just want something to work. Algolia seems to do that.
A river was just a mapping from a database to ES. For example, you could search CouchDB with ES with the proper river set up.
ES actually started with Rivers being part of Elastic but since deprecated it. You will still find people talking about "rivers" though as a description for however they are updating the index.