I was watching a show and there was a dialogue
Her blood flows through my veins.
Why do they say veins? Why not arteries? What is it's origin?
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That use of 'vein' in the dialog is figurative, and probably has this meaning:
- a. Personal character or disposition; also, a particular element or trait in this.
["vein, n.". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/221929?rskey=UAtO5O&result=1&isAdvanced=false (accessed December 09, 2015).]
With that meaning, "her blood flows in my veins" signifies that the speaker shares some particular element or elements, trait or traits, of "her" personal character or disposition.
For 'vein', and not counting obsolete or miscellaneous figurative senses, the OED lists about 7 figurative senses. Obsolete and miscellaneous figurative senses, which latter extend the breadth and depth of figurative use, bring that total of 7 to over 2 dozen.
For 'artery', however, the OED lists only 1 figurative sense:
- In fig. and in figurative contexts, esp. in the sense ‘a channel conveying life and sustenance’.
["artery, n.". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/11157?rskey=eDDyhT&result=1&isAdvanced=false (accessed December 09, 2015).]
This figurative meaning of 'artery' is unlikely to be what the speaker in the dialog intended to convey, and so the speaker chose 'veins' instead of 'arteries'.
Of course either term, in context, may carry figurative meaning as yet undreamed of by any lexicographer; the comparative numbers of figurative senses for 'vein' and 'artery' found in the OED offer nothing more than a crude indicator of the range of figurative senses for the terms.
About the etymology or origin of the word 'veins': that origin is no different for figurative uses than it is for other uses. The OED gives this etymology:
Etymology: < Old French veine, vaine (French veine ) < Latin vēna (compare vene n.), whence also Provençal vena, Spanish vena, Italian vena, Portuguese veia (†veya, vea).