All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
It is impermissible to call the created being "Ar-Rahmaan" because it is one of those names that are peculiar to Allaah.
An-Nawawi wrote, "You should know that it is prohibited to name someone with the name 'Malik Al-Amlaak' ('King of Kings'). It is also prohibited to name human beings with the names that are peculiar to Allaah, such as 'Ar-Rahmaan', 'Al-Quddoos', 'Al-Muhaymin', 'Khaaliq Al-Khalq', etc." [Sharh Muslim]
Ar-Ramli wrote, "It is deduced from their statements that it is permissible to name human beings with the names that are not peculiar to Allaah but it is prohibited to give someone any of the names and attributes that are exclusively reserved for Him, as has been stated by An-Nawawi in Sharh Saheeh Muslim." [Haashiyat Ar-Ramli ʻala Asna Al-Mataalib]
Ibn Al-Qayyim wrote:
"This means that it is impermissible to name someone with the names that are exclusive for Allaah. As for the names that are not exclusive for Him and that can be used to refer to human beings, such as 'As-Samee'' (All-Hearing), 'Al-Baseer' (All-Seeing), 'Ar-Ra'oof' (The Kind and Compassionate), and 'Ar-Raheem' (The Merciful), it is permissible to use these meanings in reference to the creation of Allaah. However, it is impermissible to use such names and attributes in reference to the creation of Allaah except when the definite article 'Al' (The) is removed from the name or attribute (because the definite article implies uniqueness and exclusivity)." [Tuhfat Al-Mawdood]
Hence, it is impermissible to call a person with the name "Rahmaan" because this name is peculiar to Allaah, The Exalted. It should be noted, though, that calling or naming someone with the names that are peculiar to Allaah is not shirk in principle unless it is accompanied by a statement, action, or belief that comprises shirk on part of the doer.
Allaah knows best.