Etymology of the word "Nazir"

Before Jacob died, he blessed his sons in (ESV) Genesis 49:

26 The blessings of your father are mighty beyond the blessings of my parents, up to the bounties of the everlasting hills. May they be on the head of Joseph, and on the brow of him who was set apart from his brothers.

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Jacob singled Joseph out among his brothers by the word
נְזִ֥יר (nə·zîr)
Noun - masculine singular construct
Strong's 5139: Nazirite -- one consecrated, devoted

The word had some nuances, Brown-Driver-Briggs:

1. of prince, ruler, as consecrated:
2. devotee (GFM), Nazirite

New International Version:

Let all these rest on the head of Joseph, on the brow of the prince among his brothers.

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Centuries later, Moses blessed the Israelites before he died in (ESV) Deuteronomy 33:

16 with the best gifts of the earth and its fullness and the favor of him who dwells in the bush. May these rest on the head of Joseph, on the pate of him who is prince among his brothers.

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This was the same Hebrew phrase used earlier by Jacob. Now ESV decided to translate it differently. That's a translation inconsistency that can be justified by the context here:

17A firstborn bull—he has majesty, and his horns are the horns of a wild ox; with them he shall gore the peoples, all of them, to the ends of the earth; they are the ten thousands of Ephraim, and they are the thousands of Manasseh.”

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Why was Nazir translated as “separated from his brothers” in Genesis but “Prince among his brothers” in Deuteronomy?

H5139 (nazir) could mean either. In the latter, ESV translators wanted to emphasize Joseph's majestic princely power.

Was the meaning of Nazir (to consecrate/set apart) etymologically related to Sar (Prince)?

Not really. It was related to H5144 (nazar), Strong's Concordance:

nazar: consecrate
Original Word: נָזַר
Part of Speech: Verb
Definition: to dedicate, consecrate

However, H5139 (nazir) itself does carry the nuance of "prince".