How did the Companions seek blessings from the Prophet (peace be upon him), and what is the difference between this seeking of blessings and shirk (idolatry)?

Seeking blessings (Tabarruk) is the act of seeking to obtain goodness and blessings by approaching something. Seeking blessings through the relics of the Prophet, peace and blessings be upon him, such as his hair and sweat, is permissible due to the blessings God has placed in them. This is not polytheism (shirk), as polytheism is the worship of other than God.

As for seeking blessings from righteous living people other than the Prophet, peace and blessings be upon him, it is an innovation (bid'ah). It can even be major polytheism (shirk akbar) if the one seeking blessings believes that the righteous person can benefit or harm, or dispose of matters in the universe.