Essential means "important to" or "a critical part of," but the word doesn't suggest in what way or order that critical or important thing is related to the subject. All things essential to a subject are of equal importance.
Fundamental is a bit more complex. It suggests "important to" and "a critical part of," but it's better to say it means "you need this before you move forward." Thus, something fundamental is essential, but that does not mean that something essential is fundamental. Mathematics are fundamental to all science classes, but a calculator isn't, though the calculator may be essential to pass the science classes.
Essence is also a bit complex. It can mean (a) an intrinsic part of, (b) is synonymous with, (c) an odor or smell, (d) the part you need to have, (e) something intrinsic to something else, and other things.
How you combine the words is important. Saying something is "fundamentally essential" is redundant, as would saying "essentially fundamental." However, you can say "fundamental essence" because "essence" can be a noun describing a smell or the most important aspect (the most essential aspect) of something. This sounds redundant, but it really isn't. Remember, something fundamental is essential, but something essential is not necessarily fundamental.
Therefore, "fundamental essence" is fine to use, because it means "the most imporant aspect of something without which that something would cease to be" (in the context you're seeing). I suspect it is rare that there is only one essential thing about any subject. For example:
The fundamental essence of a car is the engine.
There are many essential parts of a car, but what differentiates a car from a cart is the engine.
The fundamental essence of life is that we are all connected.
A great many things are essential to life: food, love, oxygen, warmth... The article you're reading is expressing the idea that the connection between people is the most imporant aspect of life. In this case, the phrase is more poetic emphasis than factual declarative (and to prove the point, I'm a natural hermit, so for me the fundamental essence of life is chocolate. Which makes perfect sense, right?)