Sense is better because it is not meaning that is meant here, as blunt as this sounds.
I would not object to "... (and not only by its geographical meaning)", I would regard it as a deliberate style decision, as sort of contrast to more common "... (and not only in the geographical sense)". Observe the change "the" to "its". "Its" is focusing on distant more accurately and is far less cold than the. It is giving a personal edge.
Meaning is for the most part of its usage technical, while in this segment the student is talking about emotional surroundings. That is one side of the story.
Another side is the fact that we are talking about ambiguity. We want to express that this usage does not have (only) its literal meaning. Since the ambiguity is a target, sense contains sufficient associative material to express it.
Third is that meaning does not go well with the preposition in. More common is with or by. However, once you try with, it does not fit the remaining structure of the sentence well.
Fourth is that meaning goes structurally well with have as a direct object. Typically, a word has this meaning. Typically, a word does not have this sense, it can be taken or understood in this or that sense. In the sentence, it is suggested that a word should not be understood by its literal meaning, in its literal sense.
Fifth is similar to what we have said already, there is "not only", so the author wants to combine two or more meanings. In order to do that you need the associative field of emotion, taste and from there sense.
Sixth, another word is calling for sense, it is distant. Distant has a similar overlap with its mathematical part as sense has with meaning. Once you employ sense, distant is immediately getting into its non-mathematical association field, you simply read it that way.
Seventh, the author is already using quite a technical term "geographical". Together with meaning it all becomes quite dissonant for the emotional intentions. It is not that dissonance is bad, it is that the structure becomes unbalanced, it sounds like an error.
Eighth, the author is using the, the geographical meaning, which is making this part cold as a stone.
From all the above, I would not say that sense is better, I would say that it is incorrect not to use it.
All in all, I would consider this usage above the suggested optional. Although a lot of people are going to object the segment for different reasons, feelings or simple expectations, I think that most of them will find at least one of the listed reasons plausible.