According to the Jewish Encyclopedia:
An agreement between two contracting parties, originally sealed with blood; a bond, or a law; a permanent religious dispensation. The old, primitive way of concluding a covenant (, "to cut a covenant") was for the covenanters to cut into each other's arm and suck the blood, the mixing of the blood rendering them "brothers of the covenant" (see Trumbull, "The Blood Covenant," pp. 5 et seq., 322; W. R. Smith, "Religion of the Semites," pp. 296 et seq., 460 et seq.; compare Herodotus, iii. 8, iv. 70). Whether "berit" is to be derived from "barah" =to cut or from a root cognate with the Assyrian "berit" = fetter (see Nathauael Schmidt, in Cheyne and Black,"Encyc. Bibl." s.v."Covenant"), or whether both Assyrian and Hebrew come from "barah"= to cut (compare "asar" = covenant and bracelet in Arabic; see Trumbull, l.c. pp. 64 et seq.), can not be decided here. A rite expressive of the same idea is (see Jer. xxxiv. 18; compare Gen. xv. et seq.) the cutting of a sacrificial animal into two parts, between which the contracting parties pass, showing thereby that they are bound to each other; the eating together of the meat, which usually follows, reiterating the same idea. Originally the covenant was a bond of life-fellowship, where the mingling of the blood was deemed essential. In the course of time aversion to imbibing human blood eliminated the sucking of the blood, and the eating and drinking together became in itself the means of covenanting, while the act was solemnized by the invocation of the Deity in an oath, or by the presence of representative symbols of the Deity, such as seven animals, or seven stones or wells, indicative of the seven astral deities; whence ("to be bound by the holy seven") as an equivalent for "swearing" in pre-Mosaic times (see Gen. xxi. 27, xxvi. 28, xxxi. 54; Herodotus, iii. 8; Josh. ix. 14; II Sam. iii. 12-20; W. R. Smith, l.c. pp. 252 et seq.). Salt was especially selected together with bread for the conclusion of a covenant (Num. xviii. 19; see W. R. Smith, l.c. p. 252; Trumbull, "The Covenant of Salt," 1899)...
The excellent article continues, discussing covenants among nations, etc.
The article suggests that the practice originally involved drinking the blood of the other covenanter by cutting into each other. If so we can see the association with "this cup is the new covenant in my blood". However, literally partaking of another's blood is, I believe forbidden:
[Act 15:20, 29 KJV] (20) But that we write unto them, that they abstain from pollutions of idols, and from fornication, and from things strangled, and from blood. ... (29) That ye abstain from meats offered to idols, and from blood, and from things strangled, and from fornication: from which if ye keep yourselves, ye shall do well. Fare ye well.
[Act 21:25 KJV] (25) As touching the Gentiles which believe, we have written and concluded that they observe no such thing, save only that they keep themselves from things offered to idols, and from blood, and from strangled, and from fornication.
Paul does use the image of a "spiritual circumcision" as "putting away the flesh":
[Col 2:11 KJV] (11) In whom also ye are circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcision of Christ:
I consider it a metaphor and not an appeal to the etymology of any other word.