The korban elevates the animal to the level of man

במד' שור או כשב כו' אדם ובהמה תושיע דין אדם ובהמה שוין מיום השמיני והלאה וביום השמיני ימול.

In the Midrash on "an ox or a sheep…" (Vayikra 22:27): "Man and beast You save, Hashem" (Tehillim 36:7) — it is the law that man and beast are equal from the eighth day onward, [as] "and on the eighth day he shall be circumcised."

The Sefas Emes opens with the Midrash linking the animal fit for korban only from its eighth day to the human bris milah on the eighth day — drawing a parallel between man and beast at the "eighth."

כי הנה סדר הבריאה החי מתבטל אל המדבר.

For behold, the order of creation is that the living animal (the "chai") is nullified to the speaking being (the "medaber").

In the hierarchy of creation, the animal kingdom exists to be subsumed within and elevated by the human being, the "speaker" who possesses da'as.

והאדם שהוא המדבר מעלה הכל אל השורש.

And the human being, who is the "speaker," raises everything up to its root.

Man's role is to lift all the lower creations back to their source in Hashem — he is the one who elevates the world.

כמ"ש שכולן נבראו לשמשני ואני לשמש את קוני.

As it is said, "they were all created to serve me, and I to serve my Maker."

The lower creations serve man, and man in turn serves Hashem — so through man's avodah the whole chain of creation is directed back toward its Creator.

א"כ הב"ח שמתקדש לקרבן על המזבח נתעלה יותר מדרך הטבע להיות קרבן לה' לכן הוא בשמיני כמו המילה בשמיני ע"י. שהוא מדרגה למעלה מן הטבע כמ"ש מי יעלה לנו השמימה.

If so, the animal (behemah) that is sanctified for a korban upon the mizbeach is elevated beyond the way of nature, to become an offering to Hashem; therefore it is [fit] on the eighth [day], like the milah on the eighth day through it — for this is a level above nature, as it is written, "Who shall ascend for us to the heavens?"

When an animal becomes a korban it is raised above its natural station. The number eight signifies the supernatural — one beyond the seven of nature/Creation — which is why both the korban and bris milah occur on the eighth day, marking an ascent above teva.

כמו כן מיום השמיני והלאה ירצה.

Likewise, "from the eighth day onward it shall be accepted (yeiratzeh)."

Only from the eighth day is the animal "accepted" as a korban — because only then can it reach the supernatural level the eighth represents.

ולכן בשחוטי חוץ כתי' דם יחשב. כשופך דם אדם.

And therefore regarding those slaughtered outside [the Mikdash] it is written, "blood shall be reckoned [to him]" — like one who sheds the blood of a man.

One who slaughters a consecrated animal outside the Mikdash is treated as a murderer ("he has shed blood"), because that animal had already been raised to the status of "adam."

לפי שכשנתקדש לקרבן נתעלה לבחי' אדם כנ"ל:

Because once it was sanctified for a korban, it was elevated to the aspect of "adam" (man), as above.

By being consecrated as a korban, the animal is lifted to the human level of "adam" — so destroying it improperly is reckoned like shedding human blood, confirming that the korban truly elevates the beast above its natural rank.

Summary: The Midrash equates man and beast "from the eighth day onward" — the animal fit for a korban only from its eighth day, like a human bris on the eighth. In the order of creation the animal is meant to be nullified to and elevated by man, the "speaker," who raises all things to their root in Hashem. When an animal is consecrated as a korban it is lifted above nature to the level of "adam" — and the number eight, one beyond the seven of nature, marks this supernatural ascent. This is why a consecrated animal slaughtered outside the Mikdash is reckoned like one who sheds human blood: having become a korban, it has been elevated to the status of "adam."