Arabic word for Exaggeration is Ghuloo. Exaggeration (in the religion) means exceeding the limits set by Allah and His Messenger ﷺ. Exaggeration, whether in belief or actions, is highly condemned in the religion of Islam.
Further Allah said: These are the limits [set by] Allah, and whoever obeys Allah and His Messenger will be admitted by Him to gardens [in Paradise] under which rivers flow, abiding eternally therein; and that is the great attainment. Quran Surah Nisa 4:13
Abdullah Ibn Abbas (radhi allahu anhu) reported that Allah’s Messenger ﷺ said: “And beware of going to extremes in religious matters, for those who came before you were destroyed because of going to extremes in religious matters.".” Sunan an-Nasa'i 3057
Exaggeration in regards to the honor and status of righteous people was the root cause behind the deviation of previous nations. Allah’s Messenger ﷺ said: “Beware of exaggeration. Your predecessors perished on account of their exaggeration.” Sunan Ibn Majah 3029
Exaggeration in praising Allah’s Messenger ﷺ involves exceeding his status and attributing Divine qualities to him ﷺ, invoking him for help, supplicating to him, swearing by him, etc. Allah’s Messenger ﷺ warned, “Do not praise me as the Christians praised the son of Maryam. Indeed, I am a slave of Allah. So say, ‘the slave of Allah and His Messenger ﷺ.’”Sahih al-Bukhari 3445 (Vol 4:654). This Hadeeth forbids exaggeration like the Christians (who exceeded the limits in praising Eesa (alaihi as-salaam)), and sufficing with the fact that Allah, the Most High, chose Muhammad ﷺ as a Prophet and a Messenger, and as His beloved.
When some people said to Allah’s Messenger ﷺ: “O Messenger of Allah, you are the best and the son of the best among us, and our master and the son of our master.” He ﷺ addressed him saying: “O people! Make your requests straightforward and let not Shaytaan lure you saying the wrong things. I am only the slave of Allah, and His Messenger. I do not like you to elevate me above my rank in which Allah, the Exalted, has praised me.” Ibne hibban 4437 and Silsilatus Sahiya Vol 4:101
At another occasion, Allah’s Messenger ﷺ interrupted small girls who described him with the attribute of the Knower of Ghayb (Unseen), ar-Rabee’ah bint Mu’awaidh (radhi allahu anhu) said: “The Messenger of Allah ﷺ came on the morning of my marriage. I had with me two servant girls, singing and mourning about my relatives who were killed on the Day of Badr, and they said while singing: “Among us there is a Prophet who knows what will happen tomorrow…” The Prophet ﷺsaid: “What are you saying? Do not say it! No one knows what will happen tomorrow except Allah, the Most Mighty and the Most Majestic.” Sunan Ibn Majah Vol. 3, Book 9, Hadith 1897. And in the narration of Saheeh al-Bukharee, he said: “Leave this, and say what you were saying (before).” Sahih al-Bukhari 5147 (Vol. 7:77)