Hi,
On my own is used in the first person (referring to myself).
In the case of "I am on my own." meaning alone / without company.
Or as a modifier to the verb "I did (something) on my own". maening I did it without help / assistance.
By yourself has the same effect but refers to a second person.
"You are by yourself". alone / without company.
"You did it by yourself". Without help / assistance.
In the first instance, both can be replaced with the adverb "alone"
"I am alone" / "Are you alone?"