Are supply and demand always (with arbitrary dimensions but still) linear? I have economics knowledge of high school level (and basic economics from my engineering degree), and I always noticed that supply and demand functions are linear (in 4+ dimensions).

It is -in practice, in real life- always the case? If not: What counterexamples (with theoretical models) do exist?
Absolutely not, linear curves are a simplification made for (many) basic economics courses. When considering the basic mechanisms of Supply and Demand (supply rises when the price rises, demand drops when the price rises, etc.) there is no need to use more complicated curves, but as soon as you introduce elements like optimization, it is in many cases necessary to use non-linear functions.

In real life we don't generally observe curves as such, but rather different quantities sold with different prices. We can estimate curves based on this kind of data, but to me that's more of an econometric exercise.