Well, since no one else has offered a thought I'll try and contribute.
Based on what you said, the problem is one that is dealt with a lot with students of mathematics. We didn't evolve with the capacity to imagine visually arbitrary dimensions. It could be that they really don't exist in any meaningful, physical way outside of mathematical curiosities. Even if we live in something like a multiverse, where there are a multiplicity of overlapping channels (like a tv that keeps going and going) that are either mostly similar, or have changing natural laws from one to the next, the 3 dimensions of average experience would have a LOT more to do with one another than they would to any of the others. If that weren't true, we would have found evidence of the others already.
Right now, if we were to discover that photons were in some way appearing to violate conservation of energy, and we looked everywhere we knew for where the energy was going and couldn't find it, we would be very curious to figure out what was going on. This sort of experiment might be an indication that conservation was happening in a more complete description of the photon, one that we currently lack, such as photons being active in places we don't yet know how to look, like a "4th" spatial dimension.
One interesting problem I started looking at, which I never even got close to in undergrad, is developing an understanding of why cross products only work in 3 dimensions, and oddly enough, in 7 dimensions(but not in 4,5, or 6). While its good that you are making an effort to think above and beyond what is required, here is another thing to ponder, what might products of 3 or more vectors in arbitrary dimensional spaces be?
However, given our understanding of vectors and how they work, and photons and how they work, we have some good reasons to expect that they are merely 3d objects. It is not merely the sensitivity of measurement, but the agreement with mathematical models that give us confidence. Based on what I suggested earlier, if photons are not 3d, they might need to be 7d, because the notion that their behavior would so closely approximate a model they have nothing to do with simply seems unlikely, and 4d photons simply might not be mathematically possible and exhibit the behavior that they do. Sometimes models can provide convenient restrictions on where to look for deeper understanding.
If you are really annoyed by an inability to quantify what vectors beyond 3d ones might be describing with those extra variables, I simply use the heuristic of telling myself each variable in the vector is some description of an object. Its possible to imagine a spectrum of colors, weights, sizes, shapes, or time as the successive variables in a vector V = i + j + k + l + m, where i j and k are position, but in order to relate this object to another one of its kind properly, you must take into consideration its color and weight, which may also change. Anyway, hope I offered some of what you might have been looking for. I mostly come here hoping to find something I can understand and learn from.