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This sounds like

something to do with the Greek pantheon.

Namely...

Zeus and his children.

My second older brother was a dangerous man, a pig who gorged himself on all our land.

This could be his son Ares, a dangerous man as God of War.

He hurt our mother’s feelings (called her a cow), so she hurt him back from belly to brow.

The mother could be Hera, who was vengeful

My oldest sister was wise, her stare — penetrating. It pinned him to the ground.

This could be his daughter Athena, Goddess of Wisdom.

He tried to say sorry but died, while my oldest brother ran on foot into the woods to hide.

This could be his son Hermes, fleet-footed Messenger God.

My father was head strong, didn’t stop to think, he insulted her too — called her a sheep.

Sounds like something Zeus would say to Hera.

My second older sister drew a sword and stabbed him in the eye (he also died).

This could be Artemis....Goddess of the Hunt.

With that in mind, there are a few thoughts I had for standing out:

Artemis, because her weapon was a bow and arrow instead of a sword;
Athena, because she was the only one not born naturally (she sprung from Zeus' head);
Ares, because of the four he was the only one born in wedlock (to Hera);
Hermes, because he's the only one whose name doesn't start with A.

There's got to be something more specific, I imagine, because I think I could form an argument for any, as above.

The oldest brother as they were the only one not to hurt anyone.

I would say

You!

Because

The story talks about how everyone in your family was violent. You always said "my oldest brother, my sister, and my father." But, you never say "I".

Answer:

The second older sister is the odd one out, she is a human wielding a sword, the rest are animals.

Reason:

Second older brother is a pig, as stated. Older sister is an owl, wise and staring. Mother is a cow/sheep. Father is headstrong like a bull or an Ox. Oldest Brother is a deer or hare, running into woods at first sign of trouble.

Is it:

The oldest brother mentioned in Line 4

Because:

Counting the syllables of each line of the puzzle, there are neat breaks within each line to separate it into two halves (Although this may be subjective). The line splits that seem intentional due to commas, '-', and the word 'and', I come to:


Line one having a split of 13 and 10 syllables

Line two is 11:10

Line 3 is 9:10

Line 4 is 8:15

Line 5 is 11:10

Line 6 is 10:10.

The only second half of each split line with a different syllable count than 10 is the 4th line, where the oldest brother runs off into the woods. This is probably quite a stretch due to some lines having multiple 'split's, such as the double commas in line 5, or the comma and '-' in line 3. In every case though where there are two possible splits, using the latter of the two will maintain the X:10 pattern, while the only line in the riddle that violates it is line 4 which only has 1 split.

I'm not too confident in this as the whole answer, but maybe it's part of the answer and will help someone finish it. Is it:

The second oldest sister

because:

All the other family members are associated with animals. The second older brother was a pig. The mother was called a cow and a sheep. The oldest sister was wise (as an owl?) The oldest brother ran into the woods to hide (what a chicken!) The father was head strong, also known as bull-headed.

Similar to my previous answer, I think that

These refer to Kronos, Rhea, and their six children Hestia, Demeter, Hera, Hades, Poseidon, and Zeus.

This is a bit more tenuous, but

Your father Kronos as God of Time is impatient; your mother Rhea is associated with the Earth (hence the cow and sheep references).

Then:

Your second oldest brother might be Hades, who's dangerous and gorges himself; Poseidon might have fled into the woods to hide because he enjoyed the company of nymphs. Your oldest sister with the death stare might be Hestia, goddesses of the hearth; and the sister with the sword might be Hera, who wasn't averse to acting out her anger.

This leaves you as

Zeus, who along with his missing sister Demeter is left out because his name doesn't start with H (I'm ignoring Kronos and Rhea purposefully.) The odd one out is therefore Poseidon, whose name does not start with H, unlike Hestia, Hera, and Hades.

Answer:

Oldest brother is the only one who didn't attack anyone in any way. (For me attack is not the same as death or threat.)

Answer:

Oldest brother: He is the only male left because he ran on foot into the woods to hide. Moreover wasn't like others who are quite aggressive.

Guess #1
My answer:

The oldest sister doesn't belong.

Because:

She is the only one with a super power! She could pin him to the ground with only a stare. ;)

Guess #2
My answer:

The oldest brother doesn't belong.

Because:

He was the only one who was a coward!

My Guess:

Oldest Brother

Reason

He was the only one not to interact with another family member.

line 3 - the oldest sister, because no rhyme