The only issue I have with it is if its a 5 man team, you bringing another 5 man team to bowl with is just a bit odd. At that point its not really a 5 man team event, but rather a 10 man team event.
Using your football reference, it would be like Green Bay bringing along Detroit to play against San Francisco in the Super Bowl. While it may not be against the rules in bowling, there definitely is something a bit odd about it.
Now, I have zero issue with all 5 people working together to make the lanes more open, thats part of the game in my eyes. However, if its a 5 man team event, your team you are bowling with should be random, not 5 other buddies you hand selected because then its not a 5 man event but a 10 man event ( for some but not for others)
Edit: I would also like to add that bowling isn't just a team sport, but also an individual one. Unlike Football, Basketball, Soccer, etc. you aren't playing those sports without your team mates. I think with bowling lots of people get caught up on the individual mindset ( aka PBA style)