Esneider, please provide context. Neither "I have lunch." or "I have my lunch." are complete sentences. The following are complete sentences:
"I have lunch at 12:00"
"I have my lunch at 12:00"
"I don't eat lunch until 12:00"
"I don't eat my lunch until 12:00"
"I have lunch with my boss every Friday"
"I have my lunch with my girlfriends every Friday" (This is not technically wrong, it's just not idiomatic).
"I have lunch with my girlfriends every Friday"
I'm sure that there are many other contexts.
Rather than stay in the office at lunchtime, I took my lunch to the park and ate it there.
It makes every sort of sense because it's exactly right.......I agree that this sentence is fine. I interpret it as "I took my packed lunch to the park" rather than "I took my luncheon (the event of having lunch) to the park".
I hope that makes some sort of sense!