What is the etymology of the phrase: on my own / on your own / on our own etc

"On my own" has been a phrase for a long time, for example:

I have suffered one on my own — A Spy on the Conjurer, 1725

Often there is erasure of the object, to prevent repetition:

Another's age, shall hasten on my own — The life of Alexander Pope, 1769

So no doubt the stand alone phrase is a growth from the erased type.

The OED has the phrase going back as far as 1404:

The Comunes desiren that the Kyng shulde leve upon his owne.

Where upon and on are synonymous in this context.

The two meanings the OED gives for the phrase are:

a. In to live on (also upon) one's own: to live on one's own resources. Obs.
b. On one's own initiative, account, etc.; (now usually) alone, unaccompanied; by oneself.

b's meaning has examples going back to 1895.

Given meaning b, on's meaning seems to have evolved from the meaning as in "on my honour", which is about hands being on a bible.

From the OED's entry for on:

With or as with the hands touching (a bible, etc.), in making an oath; using or invoking as the basis of an oath, affirmation, etc.

The idiomatic expression is from the mid 20th century:

On one's own

By one's own efforts or resources, as in:

• He built the entire addition on his own. [Mid-1900s ]

Responsible for oneself, independent of outside help or control, as in:

• Dave moved out last fall; he's on his own now. [Mid-1900s ]

(The American Heritage® Idioms Dictionary)

Early usage examples:

From Maryland farmer: (1943)

• Captain Eddie Rickenbacker says: "Once you put a boy in a foxhole, he is on his own. Once you drop a parachute trooper behind the enemy's lines, he is on his own. The youth in the cockpit of a fighter, in a tank, on a commando raid, is always ...

From On-the-farm-training for Veterans:(1947)

• ...the rest of the time he is on his own following the projects which have been assigned to him in order to make a report to the instructor when he comes the next week. Senator Morse.