PCRE doesn't implement that "restricted type of algorithm", and I think that's what the parent is suggesting is wrong, not so much the difference between formal understanding vs informal, intuition of the same subject (I would argue that the latter would be sufficient, but few even have that).

Understanding REs (a little more) formally, one can exploit their properties for e.g. performance and/or concision. For an example of what that understanding gives you, I'd recommend checking out the Ragel state machine compiler.