I live by myself: This means I live alone. I don't have a
roommate or any other family members living with me. The meaning
is very clear.
I live on my own: I found the following definitions from Cambridge.
(all) on your own
1 alone: I like living on my own.
2 without any help: I did my buttons up all on my own, Mummy.
stand on your own (two) feet INFORMAL
to be able to provide all of the things you need for living without help from anyone else:
She'll have to get a job and learn to stand on her own two feet sooner or later.
On the other hand, dictionary.com has the following meaning:
Idiom:
on (one's) own
- By one's own efforts: She got the job on her own.
- Responsible for oneself; independent of outside help or control: He is now out of college and on his own.
It could be the difference between BrE and AmE.