This is a very fascinating question, which, to my knowledge, has never received an exhaustive answer. I can share some ideas on this concern, not the definitive solution.
The general problem is that Fregean philosophic system and (post-)Saussurean Linguistics speak about two different topics. Moreover, these two sciences have different objects of analysis. What Frege had in mind was not the ordinary language — despite the fact that he made some examples from it — but, rather, the ideal language of science. The ideal language has the following properties:
- both referents (numbers, sets, truth values) and senses (mathematical and logical expressions) are abstract, in nature;
- there is no ambiguity or subjectivity (Frege explicitly states his lack of interest in the individual Vorstellungen 'mental representations'); all the reference links between singular terms and entities are always determinate (if not even rigid in Kripke's sense): e.g. the referent of the description "the positive square root of 9" will be necessarily "3", always, doubtlessly and indisputably (i.e. in all possible worlds);
- ideal language is not a system of communication; in a certain way, logic and mathematics consists of the language that they are described in, but they do not communicate it.
Note that many post-Fregean philosophers of language attempted to apply this approach to the ordinary language of the everyday communication, but these efforts have a major limit in the fact that the ordinary language is on the opposite of the ideal language for many reasons:
- linguistic expressions are still general and abstract, but referents are of a totally different nature, surely not abstract; thus, the word apple can refer to a well defined apple that I am pointing to at a certain moment, and that is totally different — in color, shape and taste — from all the other apples existing on earth;
- ordinary communication through natural language is necessarily subjective: when we want to refer to the things around us, the most effective means is to link them to our own person; thus, my bag is far more determinate and evident to my hearers than simply bag; this also means that the content that I have in mind while using words, e.g. "beautiful", "happiness" or "crime", is totally individual and can differ from someone else's understanding;
- natural language has the communication as its main aim (contra Chomsky); the grammar of a sentence is as it is mainly in order to facilitate the understanding; e.g. the topic comes first — because the hearers are supposed to know it already — while the comment comes last.
To put it simply, Fregean ideal languages lives in the "third realm" of the objective knowledge, where the reference link is determinate, while the ordinary language lives in the "first realm" of subjectivity. Therefore, Frege's notions of sense and denotation are hardly applicable to the words and expressions of the ordinary languages.
It is true that Frege made such examples as the Venus as morning/evening star paradox, but what he was doing, at that moment, was an attempt to ground the arithmetics on logic. Therefore, both senses and referents are mathematical abstractions. The phenomenon of reference, here, is a "function" from third realm to third realm. On the contrary, in the real life, the reference goes from first realm (subjectivity) to second realm (physical objects), or sometimes to third realm (when we speak about abstractions). As a consequence, the ideal language judgements are proved theorems, while ordinary language sentences are, in the best cases, only conjectures. Therefore, the ideal reference is a mathematical fact, while the ordinary reference is a matter of practical agreement. E.g., we do not know really how to define such notion as "happiness" or even "horse" or "table", but then, in an individual conversation, we try to reach an agreement with our discussants on how to indent these words in the present circumstances (but such an understanding is not established forever).
Some post-Fregean philosophers (first of all, the late Wittgenstein) have faced the indeterminacy of the ordinary language without even knowing that Saussure's idea of arbitrariness had addressed it long before.
Wittgenstein imagines the following paradox. The speaker A asks the speaker B what is sum of 5 and 8. They both agree that is is 13. Can be the speaker A certain that s/he will agree with B on every other sum? The answer seems to be negative. The private languages cannot be shared interpersonally, by definition.
Saussure was aware of this arbitrariness, and even quite unhappy because of it. But it was the Danish linguist Hjelmslev, Saussure's ideal follower, who described the ultimate consequences of the arbitrariness. Not only are words arbitrary per se (the phonological form is arbitrary with respect to the semantic content); not only the semantic boundaries among words are not universal, not given in advance, but, on the contrary, arbitrary established for each individual language; but, also the link between words and objects (i.e. referents) is arbitrary. In theory, I could use whatever word for referring whatever thing. In practice, it happens fairly often that someone uses an unexpected linguistic expression in order to refer to some real object that is not usually called that way. Still, the communication works.
Now, I think that only with these premises one can seriously consider the influence and the relevance of Frege's approach to language in contemporary linguistics.
Frege's idea of an independent realm of objective ideas (see 1918a here) has been stolen by Popper who popularized is as the "third world".
See here on Wittgenstein's rule-following paradox.
Hjelmslev's theory of arbitrariness is surveyed in his Prolegomena to a Theory of Language, cf. here.