I would like to know if there are particular signals that indicate whether a site is using Solr or another search platform.
2 Answers
A well designed system usually has a layer of abstraction which hides the backend stack. There are many types of implementations for getting a text index. Various products available are: Elastic Search, Lucene, Solr, Sphinx, Postgres GIN index(with ts_vector), etc. A good comparison is given in this link.
If you can see the javascript asynchronous call to a particular end pt. on one of the search technologies; those people are putting themselves at serious security risk.

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As commented by @kretan, no. You might be able to do a more or less qualified guess if the service exposes the query language for the backend service, but there's several services that uses the same syntax such as the Lucene query syntax - Elastic, Solr, raw Lucene, etc.).
You'll probably get better results by searching the web for the name of the company and if they've disclosed details about their platform on conferences or in technical articles (or go to a meetup and ask).

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