"we were too distant (and not only in the geographical meaning) ..."

I think the usage of meaning in the OP is right, although the sentence is flawed. The geographical meaning (i.e. definition) of distance is the distance between two points. It expresses itself in terms of metres, kilometres, feet, yards and miles. Geographically-speaking 'distance' can be measured, more or less precisely. The OP uses meaning clearly and correctly.

The problem is that the quote seems to be saying that the two people were distant from each other in ways besides the geographical meaning of the word. Distance may be 'sensed' in many ways, not all of them universally understood or acknowledged. The additional possibilities for their distance apart - emotional, financial, creative ... - are the reason why the word 'sense' seems necessary in this phrase. Precisely because their 'distance' apart is hard to define and nebulous it seems apposite to 'sense' this distance.

It is not helpful, in this context, to insist that 'meaning' often comes close to having the same meaning as 'sense'. The author seems to be wanting to highlight the other types of distance between her and her friend. It might help if the sentence were reworked so that the senses of distance, besides geographical, can be made more obvious.

For my money, I would keep '...geographical meaning' in the OP as it stands. 'Geographical sense' takes me to another place entirely - geographic sensibilities and their love of Pisgah eminences, synoptic visions and patterns at various scales. Almost certainly not what the author had in mind.

"we were too distant (and not only in the geographical meaning) ..."

I think the usage of 'meaning' in the OP is right, although the sentence is flawed. The geographical meaning (i.e. definition) of distance is the distance between two points. It expresses itself in terms of metres, kilometres, feet, yards and miles. Geographically-speaking 'distance' can be measured, more or less precisely. The OP uses meaning clearly and correctly.

The problem is that the quote seems to be saying that the two people were distant from each other in ways besides the geographical meaning of the word. Distance may be 'sensed' in many ways, not all of them universally understood or acknowledged. The additional possibilities for their distance apart - emotional, financial, creative ... - are the reason why the word 'sense' seems necessary in this phrase. Precisely because their 'distance' apart is hard to define and nebulous it seems apposite to 'sense' this distance.

It is not helpful, in this context, to insist that 'meaning' often comes close to having the same meaning as 'sense'. The author seems to be wanting to highlight the other types of distance between her and her friend. It might help if the sentence were reworked so that the senses of distance, besides geographical, can be made more obvious.

For my money, I would keep '...geographical meaning' in the OP as it stands. 'Geographical sense' takes me to another place entirely - geographic sensibilities and their love of Pisgah eminences, synoptic visions and patterns at various scales. Almost certainly not what the author had in mind.