There are a couple of points in your question to consider.
I do not understand why he cannot access the iCloud+ account.
Because you can see that your son is part of your Family Sharing group, and that he is consuming no shared space we can conclude that he is accessing your iCloud+ account.
he keeps getting messages that his storage is almost full.
From where is this message coming? It is more likely that his device (iPhone? Mac?) is alerting him to the lack of space left on the device. Storage on a device and on iCloud are different things. Apple explains this in What's the difference between device storage and iCloud storage?
Manage storage on iPhone
and
Manage your photo and video storage
have suggestions on reducing the space used on an iPhone. This is less of a problem on a Mac, though it does happen: Free up storage space on Mac
Go into his device and see what relationship it has with iCloud. On an iPhone or a Mac you can find this at Settings > [His name] > iCloud
This will open up a screen with a list of the various iCloud services he is using and the amount of space used for each of them. He will at least be using Mail and Messages.
Apple has a detailed page on iCloud use: Check your iCloud storage on any device
Once you have established this you can then look at why his account in your Family Sharing group seems to indicate no shared space being used.