Al Safi is a well-known milk company in Saudi Arabia. Milk is among the purest of things Created by Allah, and no one disputes its purity.
In the same way, the word Sufi points to the purification of the heart.
One could say, as an analogy, "Al Safi is Al Sufi." Just as Al Safi is associated with pure milk, Sufism is concerned with the inner purification of the human heart. Its goal is not merely outward appearance, but inward sincerity, love of Allah, and excellence in worship.
The purest hearts were those of the Prophets (peace be upon them). After them came those who followed their path with truthfulness and excellence. In this sense, they were the truest examples of hearts that had been purified - the true Sufis.
Look at how milk is formed...
بَيْنِ فَرْثٍ وَدَمٍ لَّبَنًا خَالِصًا سَائِغًا لِّلشَّارِبِينَ
"…between digested matter and blood, pure milk, pleasant for those who drink."
(Surah An-Nahl 16:66)
Allah produces pure milk from between digested matter and blood. Likewise, a pure heart is often formed between trials, hardships, disappointments, and tests. The difficulties a believer experiences are, by Allah's wisdom, part of the process of purification. Someone who focuses only on external purity may miss this deeper reality.
The purest things often emerge from the midst of what appears impure. So a Sufi does not simply mean withdrawing to a mountain or living in isolation. Such a person may be a Sufi - or may not be. Likewise, someone living in the middle of a crowded city, surrounded by the noise and distractions of life, may possess a heart that is completely attached to Allah. It is often said that cities like Mumbai had the most of Allah's beloved servants (Awliya) in India.
Most of us who call ourselves Sufis are simply striving to purify our hearts. It is a journey, not a claim of perfection.
Allah Azzawajal created the human heart to receive what is pure - just as a vessel receives pure milk. The heart is the place where the remembrance of our Rabb settles. When the heart is purified, it becomes filled with love, sincerity, gratitude, patience, and constant remembrance of Allah.
فَالْقَلْبُ بَيْتُ الرَّبِّ جَلَّ جَلَالُهُ
حُبًّا وَإِخْلَاصًا مَعَ الْإِحْسَانِ
"The heart is the dwelling of devotion to the Lord, Glorious and Majestic - filled with love, sincerity, and ihsan."
From the Nooniyyah of Ibn al-Qayyim al-Jawziyyah (rahimahullah):