I've been a saxophone player for more than 35 yrs and have practiced and played the sax hundreds upon hundreds of hours. From my personal experience it's good for your lungs to play saxophone. And not only for your lungs but for your entire body.
Playing Saxophone will help you develop more control over your breathing and airflow. Keeping a good posture and airflow are both fundamental for developing a better sax sound.
Good airflow on Saxophone
If you are a saxophone beginner you may find it hard to keep the air under control as you have to focus on several other things simultaneously.
First, you need to remember what keys to press on the saxophone in order to play the right notes. Secondly, you must get the saxophone reed vibrating in order to produce any sound at all.
At the same time you must keep a good Saxophone Embouchure which, along with the air flowing from your lungs and the vibration from the reed, makes up the foundation of a good saxophone tone. These are also the key areas you need to focus on to get a better sax sound as a beginner.
This extends to other woodwind and brass instruments too. If you are playing any instrument that requires you to blow air into a mouthpiece you'll find that having a good airflow will make a huge difference in your performance when it comes to sound and endurance.
There's actually a little hack that you can use in order to help you use more of your lungs than you normally would when you're not playing saxophone.
Improving your airflow on Saxophone
When you stand straight up (or sit up with a straight back) you are making it easier for the air to flow from your lungs to your saxophone through the mouthpiece.
Imagine you have an invisible string attached to your head and that it is pulling you up, keeping your back straight and pushing the chest forward as if you just won a medal that you want to show off.
Simply by keeping that mental picture in your mind you'll notice that your airways are more open which also improves your saxophone tone. Pretty cool, right?!
Try this infront of a mirror, so that you can take note of any adjustments you need to make to the neck strap and mouthpiece in order to stand (or sit) straight up.
To get back to the main question about if playing the sax is good for your lungs or not I'd say that it is. And it's lots of fun playing sax too!
Check out the article “7 tips for a better saxophone sound” if you want more tips on how to improve your saxophone tone and develop more control over your breathing, your lungs and diaphragm.