This article misleads you by conflating regular expressions with specific implementations like PCRE, which also does non-regex string matches. Annoyingly, the article does a good job of explaining what a regex is and what the limitations of regex are relative to PCRE, so the author should understand that what they are talking about when they talk about NP-complete string matching is not regex, but PCRE-specific features.

The distinction matters because regex absolutely can't match HTML, and because regex, unlike PCRE expressions, have guaranteed O(1) space and O(n) time complexity when matching a string of length n. When you use PCRE features for string matching, that may degrade to exponential time which makes it useless. For example, you can do denial of service PCRE attacks, but not denial of service regex attacks (unless you can query with some megabyte-large regex).