PostgreSQL PostGIS vs ElasticSearch Spatial search
The postgis scales the same way as postgresql scales. The postgis index will work pretty the same as other relational, you may check it out here.
If you take a look at the link, it explains that it indexes using some geometrical algorithm which is performed on each insert operation, so it might not be responsive enough in real-time apps.
While elasticsearch have real time indexing, based on Lucene index. Elastic search generally suits better for realtime heavy applications, then Postgresql.
Postgresql has a huge advantage it is simplicity. It is much easier to implement test and maintain such a feature using Postgresql. For example, I prefer creating prototype based on Postgresql quickly, and if it starts perform bad because of big amount of writes, etc. I switch to elasticsearch implementation.