"Welcome to my world" is a somewhat sarcastic way to say "Yeah, that happens to me as well".
"Welcome to my world" could be said with good meaning. For example:
Mum 1: "I got an hour's sleep last night. I am exhausted".
Mum 2: "Welcome to my world!"
In that case, Mum 2 could be saying "You're not alone. I understand you". But, more often, the message that's actually being sent is "Oh, stop whining. You're not the only person in the world that's feeling tired!".
It could be even more 'aggressive' in how it's said, as well.
Employee: "I'm sorry I didn't get that work done. I had so much to do last night, I just couldn't find the time to do it!".
Manager: "Welcome to my world!"
In that case, it might mean "Yeah, well, you're not the only one that is busy. Everyone else manages to find time to do the work around their other commitments".