Generalising a bit, Solr is more targetted at enterprise search and unstructured content search (e.g. bundled with most content management systems), and Elasticsearch is more targetted at data analytics and structured data search (e.g. with the ELK stack). Again simplifying a bit, Solr can be a bit more configurable and/or work better with the types of data that benefit from more configuration, and Elasticsearch can work better with the types of data that work more "out of the box".
I'd agree that there don't seem to be a huge number of openings for specialist search roles or a huge number of people specialising in search, but it is often part of another role and there are often people who have touched on search in their roles. That suggests that many people are just using it with largely default setups. Having said that, things like advanced relevancy tuning, if you need it, is a very much under appreciated skillset, and definitely needs someone with good experience or ability to learn.