You wrote: Furthermore, ‘the morning star’ and ‘der Morgenstern’ are distinct as signs—one is English, the other German—and each will be related to a different Begriff [concept] in the mind of each speaker who uses these terms.

I think that you have confused concepts with ideas here.

What Frege says in Ãœber Sinn und Bedeutung is: “The Bedeutung and sense of a sign are to be distinguished from the associated idea. … The idea is subjective: one man’s idea is not that of another. … This constitutes an essential distinction between the idea and the sign’s sense, which may be the common property of many people, and so is not a part or a mode of the individual mind.” The German word used for “idea” here is not Begriff, but Vorstellung.

Frege makes clear, from the Introduction of the Foundations onward, that ideas must be distinguished sharply from concepts. Not to distinguish them violates the fundamental principle always to separate sharply the psychological from the logical, the subjective from the objective. Thus: “In compliance with the first principle, I have used the word ‘idea’ [Vorstellung] always in the psychological sense, and have distinguished ideas from concepts [Begriffe] and objects [Gegenständen].” Concepts are logical and objective. The same concept-word in two different people’s mouths will designate the same concept, but will necessarily be associated with numerically different ideas.

In the case of “the morning star” and “der Morgenstern,” there is the further difficulty that “the morning star” and “der Morgenstern” are both (by Frege’s lights) proper names, not concept-words. They have ideas directly associated with them, but not concepts. (They name objects; concepts are the Bedeutungen of concept-words such as “( ) is a horse” or “( ) ist ein Pferd.”)

There may be concepts involved somewhere in the process: for example, concepts might be constitutive parts of their sense. (E.g., “the morning star” may have the sense, “the extension of the concept ‘( ) is a star that appears in the morning in such-and-such a way.’”) But if that’s the story, then “der Morgenstern” and “the morning star” must have identical concepts associated with them in order to have the same sense. (If the sense of each was given in terms of a different concept, then they would have different senses, although they could still have the same Bedeutung.)

Broadly speaking, since for Frege concepts are logical and objective, there is no reason to think that different people couldn’t express the same concept with two different signs. Indeed, I’d wager that he’d think the preservation of the concept is the best way to determine, for example, whether “( ) is a horse” is or is not an adequate translation (for logical purposes) of “( ) ist ein Pferd.”

This may seem nitpicky, but given the extreme importance that Frege puts on his notion of the Begriff, and given how emphatic he is about the distinction between that notion and ideas, and the objective and logical nature of Begriffe in particular, it’s a nit that I think he would rather want picked. (Among other things, the distinction is central to the confusion that he attributes to Benno Kerry in “On Concept and Object.”)