"A mere electric animal, and me earning all that I've worked my way up to through the years" meaning
There are two parts that make me confused about the meaning of that sentence.
1. "me earning". Does it mean "I earned"?
2. "my way up to". Does it mean "really hard"?
So 1 and 2 combined, Does that sentence mean that "I have earned all because I have worked really hard through the years (only to buy a mere electric sheep)"?

Question

"A mere electric animal, and me earning all that I've worked my way up to through the years" meaning
There are two parts that make me confused about the meaning of that sentence.
1. "me earning". Does it mean  "I earned"?
2. "my way up to". Does it mean  "really hard"?
So 1 and 2 combined, Does that sentence mean that "I have earned all because I have worked really hard through the years (only to buy a mere electric sheep)"?

Answer

In this context, 'me earning' means 'me deserving'.
'I've worked my way up to'  means  I started at a low position and gradually I've earned/worked promotions to where I am now.

The sentence is saying, he deserves all that he has now. He has worked hard to get to the position he is in now.