As essential and fundamental seems to be synonym I doubt whether I can use "the fundamental essence" expression. However I see its usage in the internet. https://daveursillo.com/the-fundamental-essence-of-life-an-incredible-truth/
Does it enhance or on the opposite side damage my writing competency?
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?)
For use in advertising or other marketing, it sounds just fine, because we've come to expect pleasant sounding phrases even when nonsensical.
In the same way quasi-philosophical discussions are full of pleasant-sounding-but-generally-meaningless phrases that capture the imagination of the reader and suggest deeper insight. When dealing with an audience who is used to this kind of verbiage, again, there is no problem.
It's only problematic if you are writing for a group of critical readers, inclined to analyze every phrase for coherent meaning. When subject to close analysis the phrase "fundamental essence" is redundant, because by definition:
essence (n): The intrinsic nature or indispensable quality of something, especially something abstract, which determines its character.
i.e. if that nature or quality was not fundamental to that thing, it would not represent its essence.
To sum up: Does it sound nice? Yes. Is it meaningful? Not really -- but lack of cogency hasn't stopped legions of writers from employing similar phrases.
