My son was like yours - a water baby! If he likes to compete and has that drive, then a swimming club will be good for him. If he is not a fiercely competitive child, then a swimming club will be fun for a couple of years until the other kids catch up. I have two swimmers - No. 1 son has been swimming since 5, joined a club at 8 when he reached level 10, and is still with the same club at 13, swims at county/regional standard. No2 son did the same, but has given up, now aged 10, as he was just not a natural swimmer (completely different body shape).
Best advice: look around at local clubs - you want one that will help your child develop all his strokes, but not one that pushes and pushes - he has several years before he can compete and so look for a club that has flexibility in training for the future and a good reputation for developing stroke technique.
Not all clubs are looking for morning swimmers, especially not at this age. ....our club does early mornings and evenings, the regional development squad for example has to swim 6x a week, but have 10 sessions to choose from. You need this level of flexibility so that you can do other things too - cubs, football, cricket etc. Until our kids get to 10ish, they only swim up to 3 x a week for an hour to an hour and a half. Most swim twice a week.
Swimming is a great sport, encourages discipline and encourages kids to compete as much against themselves and their personal bests, as with others. its very social. Kids who swim tend to have good time management skills - if they want to train, they have to get their homework done.
A swimming club will take over your life too, so be prepared for many hours sat by a hugely hot pool.