Perplexity's utility as an answer engine is RAG (retrieval augmented generation). In response to your question, they search the web, crawl relevant URLs and summarize them. They do include citations in their response to the user, but in practice no one clicks through on the tiny (1), (2) links to go to the source. So if you are one of those sources, you lose out on traffic that you would otherwise get in the old model from say a Google or Bing. When Perplexity crawls your web page in this context, they are hiding their identity according to OP, and there seems to be no way for publishers to opt out of this.
It is possible that when they crawl the web for the second use case -- to collect data for training their model -- they use the right user agent and identify themselves. A publisher may be OK with allowing their data to be crawled for use in training a model, because that use case does not directly "steal" any traffic.