The Wikipedia article referenced by the OP provides a summary of Henry Watson Fowler's analysis of the the phrase "the exception proves the rule" as it is used in English in Modern English Usage.
Fowler traces the original meaning of the phrase to a defense by Cicero. Here is how the Wikipedia article describes that original and most correct use of the phrase according to Fowler:
The phrase is derived from a legal principle of republican Rome: exceptio probat regulam in casibus non exceptis ("the exception confirms the rule in cases not excepted"), a concept first proposed by Cicero in his defence of Lucius Cornelius Balbus. This means a stated exception implies the existence of a rule to which it is the exception. The second part of Cicero's phrase, "in casibus non exceptis" or "in cases not excepted," is almost always missing from modern uses of the statement that "the exception proves the rule," which may contribute to frequent confusion and misuse of the phrase.
The exception shows the existence of a rule for all cases not excepted.
Fowler objected to the use of the phrase when it suggested the following:
- Exceptions can always be neglected.
- A truth is all the truer if it is sometimes false.
This appears to be what the OP is objecting to as well when OP asks:
Is there a logical/scientific basis for this claim? In this sense, is it truly valid in argumentation, logical thinking, and scientific enquiry?
Fowler would agree that there is no basis for this claim if it means that exceptions can be neglected or that the truth is always truer if it is sometimes false.
If one considers the use of falsifiability as a criteria for science, identifying possible exceptions to a theory, exceptions that have not yet been observed, would discredit a theory requiring that a revision to the theory be made. As Wikipedia puts it:
A statement, hypothesis, or theory has falsifiability (or is falsifiable) if it is contradicted by a basic statement, which, in an eventual successful or failed falsification, must respectively correspond to a true or hypothetical observation.
These two approaches to exceptions are different. In Cicero's case, the actual existence of the exception proved the existence of a rule in all cases not excepted. A theory is considered falsifiable, on the other hand, if one can identify potential exceptions, that may not ever be found, to show whether a theory needs to be modified or rejected.
"Exception proves the rule" Wikipedia https://en.wikipedia.org/wiki/Exception_that_proves_the_rule
"Falsifiability" Wikipedia https://en.wikipedia.org/wiki/Falsifiability