Is the aforementioned Hadith: "Choose for your daughters as you choose for your sons," a Prophetic Hadith or a saying of the Companions?

The daughter is a trust around her father's neck, and choosing a righteous husband for her is among the greatest ways a guardian can protect his daughter. The Quran, Sunnah (definition), and the biographies of the righteous predecessors contain examples of a guardian offering his female ward to a righteous man, such as the story of Musa (peace be upon him) with the righteous man, 'Umar ibn al-Khattab offering his daughter Hafsah to 'Uthman and then to Abu Bakr, and Sa'id ibn al-Musayyab marrying his daughter to his poor student. These stories demonstrate that there is no harm in this, and modesty does not prevent the guardian from seeking a righteous husband for his daughter.

As for the circulated saying: "Choose for your daughters as you choose for your sons," it is not a prophetic Hadith (definition), and whoever attributes it to the Prophet (peace and blessings be upon him) has erred.