Why do "do you feel like you're" and "does it feel like you're" mean the same thing?

I don't think they mean the same thing at all.

Does it feel like you're dying?

Here, "it" refers to a sensation of some sort, whether by internal or external stimulus.

Do you feel like you're dying?

This can refer to a state of being, either physical or mental.

Say you're in the hospital. You've been given a sedative. It makes you feel poorly.

"Does it feel like you're dying?" can refer to the effect of the sedative -- the implication is that the sedative and its effect in the body is the thing that is providing the person with the feeling as though they're dying.

"Do you feel like you're dying?" on the other hand makes no such implication -- here, the question is asking about the current state of miasma the person is experiencing, not about the effect the specific thing is having on them.

That you can ask this question out of the blue to a person and have it imply that the person could feel like death or dying does not necessarily mean that there are not times where you may choose to say one versus the other. It depends a bit on what context this comes under.

EDIT: OP requested examples.

But they don't necessarily mean the same thing. There can be subtle, although quite distinct, difference.

They could be interpreted the same way, but there are certainly contexts where they convey different meanings.

This is very much to do with the possible ambiguity of the word feel.

What follows are two possible interpretations, each using a different sense from Merriam-Webster.

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1 a : to receive or be able to receive a tactile sensation
// lost the ability to feel in his fingertips

Here is a possible context:

"I understand you just had a root canal. I can't imagine what that would feel like. Does it feel like you're dying?"

This is a figurative comparison between the experience of having a root canal and the imagined suffering of dying.

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2 b : to have a marked sentiment or opinion
// feels strongly about it

Here is a possible context:

"The doctor says you're having a heart attack. Hold my hand! Do you feel like you're dying? Is there somebody I should call?"

This is a literal question, where the person is being asked if they believe they are actually about to die.