No, your definition of strict is very fuzzy.

> For example, the string ' Daddy' matches

Are the strings given or could they come from a defined set? Like [a-zA-Z]{1,20} ? Is the set infinite?

> anything allowing classifying regexes

Please be aware that regexes are almost a complete sub-language with recursions and more. You don't expect us to solve the halting problem for the general case , don't you?

So which restricted regex syntax is allowed?

I have an idea for a clear definition of "strictness" and an approach to solve this by estimating the cardinality of possible solutions...

But I'm reluctant to spend efforts w/o a clear problem description. Sorry!

Like davido already asked, could you please clarify what for you're wanting this?

Seems pretty much to be a XY Problem for me...