He joined the company on the same day as me.

[tufguy]

1) He joined the company on the same day as me.

2) I don't know the name of the person who joined the company with me.

Please check my sentence.

 

[Amelot]

I am not a teacher.

Your first sentence is fine.

Considered in isolation, I can't tell what the second sentence is supposed to mean.

 

[GoesStation]

1) He joined the company on the same day as me.

2) I don't know the name of the person who joined the company with me.

They are both fine. Your teachers may tell you that you need I rather than me in sentence 1. Native speakers almost all say ​me.