Can one ever be understood? When people say “yeah, I feel you” do they really? Is language enough of an outlet to transmit feelings with enough exactitude?
This is akin to a major question in philosophy that is still being debated called "The Problem of Other Minds". The two broad questions within this debate are:
- The Thick Question: How can we know, be justified in claiming, or understand that other minds exist?
- The Thin Question: How can we know, be justified in claiming, or understand facts about what the other mind is thinking? (Your characterisation is closer to the thin question)
Trying to summarise every philosophical argument ever made with regards to this question is a careers worth of work, but here are some historically important positions.
- Analogy: Thinkers like Mill would say that we make claims about other minds based on our knowledge of our own. We feel pain, and so we infer that others must also feel pain.
- Phenomenology and Perception: Modern thinkers like McDowell and Dretske, as well as mid-20th century thinkers like Sartre, would claim that we perceive other minds when we see other people. Here, they would claim that when we see someone wince in pain, we are seeing the pain on their face. We aren't inferring the pain, we don't have to think about the expression and then interpret it, we just see the pain. (This is a bit strange at first glance, but it does have more appeal when you read through the arguments more. For the modern approach, McDowell is the best person to read more of. For the continental approach, try Being and Nothingness.
- Other Continental Approaches: Beyond the perception argument, existentialist thinkers also posit that other minds are built into the structure of our existence. Heidegger has Mitsein, Sartre has the Other, etc.
- Biting the Bullet: Finally, some thinkers will bite the bullet and embrace a position like solipsism. The idea being that knowledge of any minds outside our own mind is impossible. However, there are different flavours here. There will be hardcore solipsistics, who just say that and stop. Then, there are certain sceptical positions, probably Cartesian ones (though Descartes does not clearly address this) that suggest that other minds probably exist (for one or more of the reasons above) but we can't be sure.
This is a rich debate, and other important thinkers here with detailed ideas about the problem and possible solutions include giants such as Dennett, Wittgenstein, and Augustine.
Another possible interpretation of your question can be about the specific phenomenology of experience. So, can we communicate the knowledge of how an emotion "feels" through language?
While the Problem of Other Minds tangentially looks at this, this formulation is closer to question of Qualia.
Qualia is the distinctive unit of experience. The Knowledge Argument is that no amount of knowledge about the physical facts of an experience will be enough to communicate the "qualia" of that experience.
The famous argument here is the Mary argument by Frank Jackson, that argues that the "qualia" cannot be transmitted this way. I'd start there.
Jackson's argument basically comes from a thought experiment. Imagine a woman named Mary who spends her life in a black and white room with a black and white TV. In time, Mary becomes a neurophysiologist who knows all the physical facts there are to know about colour. She knows wavelengths, she knows how our eyes react to colour, etc etc. Then, one day, she is allowed to leave the room and sees the sky for the first time. Has Mary learned something new about the colour blue?
It seems intuitive to say she has, despite the fact that she knew all the physical facts that language could convey. This is the Knowledge Argument.
This will generate to the idea that when someone says they know how we feel, they cannot possibly know how we feel. We can't communicate it to them. No one can.
There are arguments against the Knowledge Claim and different flavours of it. Lewis' Ability Hypothesis and Conee's Acquaintance Hypothesis are two flavours worth looking at. Dennett and Chalmers have some interesting points as well.