As typically defined academically, yes. In the real world things might be more complex, but I think it would hold true for many implementations.

Academically a regex is a string of of:

- character literals

- "epsilon" characters (meaning empty string)

- "+" characters (which means either what we see before the + or what we see after the +, in real-world regex these are usually represented as | instead of + as well as used implicitly in constructs like [a-zA-Z])

- "*" characters (same meaning as in real-world regex)

- And parentheses to allow controlling the order of operations

In other words, "if we build a state machine which is different from a regex, it will perform differently than a regex would." That's trivially true, but is it interesting beyond restating "there is no Silver Bullet"?

Technically we aren't discussing the regex itself here but the syntax for defining a regex.

You could use a different syntax for defining the same regex, and that syntax could have different properties. For instance you could use parantheses like racket where "(...)" and "[...]" have the same meaning, but "(...]" is not well formed syntax (if I remember my racket correctly). Using such a syntax to define regex a counter would no longer suffice to decide if a string is a regex.

You could have a stack of (paren-type, integer) pairs. If the paren has a different type than the top pair on the stack (or the stack is empty), push a new pair with the integer set to one. If the integer goes to zero, pop the pair. At the end of the expression, you have to have an empty stack.