Why is it such a big deal to Daenerys Targaryen that Jon is the true heir?

Question

Ever since she found out that Jon is the true Targaryen heir to the throne, Daenerys has been worried about it, and has been wanting him to keep it a secret, despite him several times saying that he doesn't want it and considers her to be his queen. She also considers him telling his family of his past a "betrayal", and burns Varys alive for spreading that information. Ultimately it also seems a factor in her decision to go batshit insane during the attack on King's Landing.

Why is this such a big deal to her? If Jon doesn't want the throne, then how could this possibly hurt her? If the "people" prefer and love Jon over Daenerys (which doesn't make much sense, since Jon is only loved in the North and there are 6 other Kingdoms in Westeros that don't give a crap about him), then okay, that's not optimal, but if Jon doesn't want it and himself supports Daenerys, what's the issue then? Why doesn't Daenerys just start ruling and showing her capabilities as a ruler? What exactly does she fear? That Sansa will tape Jon to the Iron Throne and force him to be king?

Answer 1

Daenerys' claim to the Iron Throne is entirely based on the idea that it was stolen from her father, Aerys Targaryen, during Robert Baratheon's rebellion. Daenerys is his heir (for most of the story), and thus the "rightful" ruler of Westeros.

Jon's claim upsets that. Now she cannot justify her conquest of Westeros by claiming to be Aerys' heir.

Answer 2

She does not fear Jon claiming the throne, but the people of Westeros forcing him to. She knows that the people of Westeros don't love her like as much as they like Jon. She fears that they might do a rebellion against her to get Jon on the throne. In S08E05 Dany says to Jon:

Now she [Sansa] knows what happens when people hear the truth about you. Far more people in Westeros love you than love me. I don't have love here. I only have fear.

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Answer 3

In order to successfully rule, Daenerys needs to have control of all of the seven Kingdoms. There are many higher and lesser noble houses in Westeros and to secure the peace, she needs them to follow her. She can either achieve that by securing them as allies peacefully or by her coercing them with force.

Jon has earned a loyal followership in the North by growing up there, securing its independence in the Battle of the Bastards and his role in defeating the White Walkers. He was made King in the North by his people without even wanting the job. His people respect and admire him. Since the Starks have family bonds with the houses ruling the Vale and the Riverlands, you can expect those lands to choose him over Dany should it be necessary.

Daenerys does not have this kind of backing by the noble houses in Westeros. Her claim to the throne rests on her being its legitimate heir and her military strength (Drogon being its biggest asset). If it comes out that Jon is actually the legitimate heir to the throne, Jon has the followership and the legitimacy, while she is only left with her military strength. It is very likely that the other noble houses turn against her and she can only secure her rule through another war, which she is not guaranteed to win (even with her dragon).