What is the evidence for dividing Tawhid into three categories?

This division is taken from study and analysis. When the scholars analyzed the texts of the Book of Allah and the Sunnah of His Messenger (peace and blessings of Allah be upon him), this became clear to them. Some of them added a fourth category, which is Tahweed al-Mutaaba’ah (oneness of following). All of that is derived from study and analysis. 

Undoubtedly the one who ponders the meanings of the Quran will find that it contains verses which enjoin devoting worship sincerely to Allah alone. This is Tawhid al-Uloohiyyah (Oneness of the Divine Nature). And he will find verses which indicate that Allah is the Creator and the Provider, that He is in control of all things. This is Tawhid al-Ruboobiyyah (oneness of the Divine Lordship), which the mushrikoon affirmed, but that did not make them Muslims. He will also find other verses which indicate that Allah has the Most Beautiful Names and the Most Sublime Attributes, and that He has no equal or peer. This is Tawhid al-Asma’ wa’l-Sifaat (Oneness of the Divine Names and Attributes), which was denied by the innovators such as the Jahamiyyah, Mu’tazilah, anthropomorphists and those who followed in their footsteps. 

There are verses which indicate that it is obligatory to follow the Messenger  (peace and blessings of Allah be upon him) and to reject whatever goes against his laws. This is Tawhid al-Mutaaba’ah (Oneness of Following). This categorization is known from study and analysis of the relevant aayahs and the Sunnah. For example, Allah says (interpretation of the meaning): 

“You (Alone) we worship, and You (Alone) we ask for help (for each and everything)” [Al-Faatihah 1:5]

“O mankind! Worship your Lord (Allah), Who created you and those who were before you so that you may become Al-Muttaqoon (the pious)” [Al-Baqarah 2:21] 

“And your Ilaah (God) is One Ilaah (God — Allah), Laa Ilaaha illa Huwa (there is none who has the right to be worshipped but He), the Most Gracious, the Most Merciful”[Al-Baqarah 2:163]

“And I (Allah) created not the jinn and mankind except that they should worship Me (Alone).

I seek not any provision from them (i.e. provision for themselves or for My creatures) nor do I ask that they should feed Me (i.e. feed themselves or My creatures).

Verily, Allah is the All-Provider, Owner of Power, the Most Strong” [Al-Dhaariyaat 51:56-58] 

“Indeed, your Lord is Allah, Who created the heavens and the earth in Six Days, and then He rose over (Istawa) the Throne (really in a manner that suits His Majesty). He brings the night as a cover over the day, seeking it rapidly, and (He created) the sun, the moon, the stars subjected to His Command. Surely, His is the creation and commandment. Blessed is Allah, the Lord of the ‘Aalameen (mankind, jinn and all that exists)!” [Al-A’raaf 7:54]

“Say (O Muhammad): “He is Allah, (the) One.

 Allah-us-Samad [Allah — the Self-Sufficient Master, Whom all creatures need, (He neither eats nor drinks)].

He begets not, nor was He begotten.

And there is none co-equal or comparable unto Him” [Al-Ikhlaas 112:1-4]

“Say (O Muhammad to mankind): If you (really) love Allah, then follow me (i.e. accept Islamic Monotheism, follow the Quran and the Sunnah), Allah will love you and forgive you your sins” [Aal ‘Imraan 3:31]

“Say: “Obey Allah and obey the Messenger, but if you turn away, he (Messenger Muhammad) is only responsible for the duty placed on him (i.e. to convey Allah’s Message) and you for that placed on you. If you obey him, you shall be on the right guidance. The Messenger’s duty is only to convey (the message) in a clear way (i.e. to preach in a plain way)” [Al-Noor 24:54]

And there are many aayahs which point towards this categorization. 

In the ahadith: 

The Prophet (peace and blessings of Allah be upon him) said, according to the hadeeth of Mu’aadh whose authenticity is agreed upon (al-Bukhari, 2856; Muslim, 30): 

“It is the right of Allah upon His slaves that they should worship Him and not associate anything with Him.” 

And he (peace and blessings of Allah be upon him) said: “Whoever dies invoking anything other than Allah as a rival to Allah will enter Hell.” (Narrated by al-Bukhari, 4497; Muslim, 92). 

And he (peace and blessings of Allah be upon him) said to Jibreel (peace be upon him), when he asked him about Islam: “It means to worship Allah and not associate anything with Him, to pray the prescribed prayers, to pay the obligatory zakaah…” (Agreed upon; al-Bukhari, 50; Muslim, 9). 

And he (peace and blessings of Allah be upon him) said: “Whoever obeys me obeys Allah, and whoever disobeys me disobeys Allah.” (Agreed upon; al-Bukhari, 2957; Muslim, 1835). 

And he (peace and blessings of Allah be upon him) said: “All of my ummah will enter Paradise except those who refuse.” It was said, “O Messenger of Allah, who are those who refuse?” He said, “Whoever obeys me will enter Paradise and whoever disobeys me has refused.” (Narrated by al-Bukhari in his Saheeh, 7280). 

And there are many similar ahadith. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: “God is the One Who is worshipped and obeyed. God is the One Who is loved, and the One Who is loved is the One Who deserves to be worshipped. The reason why He deserves to be worshipped is because of His Attributes which imply that He is the Most Beloved to Whom the utmost submission is due.” 

And he said: “God is the One Who is loved and worshipped, Whom hearts love, to Whom they submit, before Whom they humble themselves, Whom they fear and hope, to Whom they turn at times of difficulty, calling upon Him with regard to their concerns, putting their trust in Him with regard to their interests, seeking refuge in Him, finding tranquility in the remembrance of Him and peace of mind in love for Him. That is only for Allah, therefore the phrase Laa ilaaha ill-Allah (there is no god except Allah) is the most truthful of speech. Those who believe this are the people of Allah and His party, and those who deny it are His enemies who are subject to His wrath and vengeance. If a person gets this right his entire situation will be good, and if he fails to achieve this then his knowledge and actions will inevitably be corrupt.” 

We ask Allah to help all the Muslims, rulers and ruled alike, to understand His religion, to adhere steadfastly to it, to be sincere towards Allah and His slaves and to beware of that which Allah has forbidden, for He is Able to do that.  May Allah bless our Prophet Muhammad and all his family and companions.