The right of legislating belongs solely to Allah alone.
Legislation is not but for Allah. He has commanded that you worship not except Him. (Qur'an 12:40)
And He shares not His legislation with anyone. (Qur'an 18:26)
In the recitation of Ibn ‘Amir, one of the seven reciters, it comes in the form of a forbiddance: {Do not commit shirk with anyone in His legislation.}
The Ayat singling 'Ibadah for Allah and negating any associates with Allah in 'Ibadah come in the same wording as singling Hakimiyyah to Allah. Just like Tawhid is to assign worship to Allah alone, and the opposite of this is Shirk, likewise Tawhid is to assign Hukm only to Allah and deviating from this is Shirk in the Hakimiyyah of Allah.
As for democracy, it's a system that grants people the right of legislating contrary to the Shari'ah (in opposition to Tawhid). In democracy, if the majority decide to make Khamr legal they are permitted do so, and if they decide to make polygyny illegal they may do that. As such, this system makes people legislators besides Allah. That's why it is Shirk. In Islam however, legislation is bound to the Shari'ah of Allah, and the people can only implement the judgement of Allah and His Messenger, and this is the essence (Asl) of Iman, as Allah said in the Qur'an:
But no, by your Lord, they will not [truly] believe until they make you, [O Muhammad], judge concerning that over which they dispute among themselves and then find within themselves no discomfort from what you have judged and submit in [full, willing] submission. (Qur'an 4:65)
There is no doubt that democracy constitutes denial of the Kitab and Sunnah in every way. Allah said in the Qur'an that:
It is not for a believing man or a believing woman, when Allah and His Messenger have decided a matter, that they should [thereafter] have any choice about their affair. And whoever disobeys Allah and His Messenger has certainly strayed into clear error. (Qur'an 33:36)
In democracy, people have a choice in affairs which Allah and His Messenger have already established.
Allah says in the Qur'an:
O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah and the Messenger, if you should believe in Allah and the Last Day. That is the best [way] and best in result. (Qur'an 4:59)
In democracy, if people disagree over anything, they may resolve such disputes by putting them up to vote, and then enact the decision of the majority. Etc.
Voters disbelieve due to,
Aiding in their Shirk, I.e. without the people voting, there is no Democracy.
Aiding in Shirk constitutes Kufr because it amounts to taking the Mushrikin as awliyah which is Kufr in its Asl, i.e. generally wala'a to the Kuffar = Kufr.
Any type of assistance for the Kuffar amounts to Kufr unless proven otherwise by Wahi or Ijma.
"O you who have believed, do not take the Jews and the Christians as allies. They are [in fact] allies of one another. And whoever is an ally to them among you - then indeed, he is [one] of them. Indeed, Allāh guides not the wrongdoing people." (Qur'an 5:51)
Voters approve their leaders judgement and this is Kufr; via this, they have taken their candidates/leaders as Lords besides Allah.
This is the exact reason some of the Jews of Bani Israel disbelieved:
At-Taubah 9:31
ٱتَّخَذُوٓا۟ أَحْبَارَهُمْ وَرُهْبَٰنَهُمْ أَرْبَابًا مِّن دُونِ ٱللَّهِ وَٱلْمَسِيحَ ٱبْنَ مَرْيَمَ وَمَآ أُمِرُوٓا۟ إِلَّا لِيَعْبُدُوٓا۟ إِلَٰهًا وَٰحِدًاۖ لَّآ إِلَٰهَ إِلَّا هُوَۚ سُبْحَٰنَهُۥ عَمَّا يُشْرِكُونَ
English - Sahih International
They have taken their scholars and monks as lords besides Allāh, and [also] the Messiah, the son of Mary. And they were not commanded except to worship one God; there is no deity except Him. Exalted is He above whatever they associate with Him.
Imam Ibn Kathir رحمه الله says in his Tafsir:
اتَّخَذُواْ أَحْبَارَهُمْ وَرُهْبَانَهُمْ أَرْبَابًا مِّن دُونِ اللّهِ
They took their rabbis and their monks to be their lords besides Allah.
Adi commented, "I said, `They did not worship them."'
The Prophet said,
بَلَى إِنَّهُمْ حَرَّمُوا عَلَيْهِمُ الْحَلَلَ وَأَحَلُّوا لَهُمُ الْحَرَامَ فَاتَّبَعُوهُمْ فَذَلِكَ عِبَادَتُهُمْ إِيَّاهُم
Yes they did. They (rabbis and monks) prohibited the allowed for them (Christians and Jews) and allowed the prohibited, and they obeyed them. This is how they worshipped them.
Shaykh 'Ali al-Khudayr's response to a few misconceptions concerning Democracy:
https://gamefaqs.gamespot.com/boards/563727-prince-of-persia/62600641
والله ولي التوفيق