Yes, HTML is not a regular language, and so cannot be matched with true regular expressions. As mentioned elsewhere, most "regex" libraries in existence today are not limited to simple regular expressions -- they simply wouldn't be powerful enough for many common tasks otherwise.

HTML is more accurately a context-free language (CFL). A regular expression does not allow you to do any sort of counting or stack-based matching in a match, which is required to do things like "I just saw <div>...<a>...<img> so I better see "</img>...</a>...</div>" later on.

The reason the linked solution works is because of the extended capabilities of PCREs, such as backtracking and things like that.