does a family member's photos taken on their iPhone load to the iCloud ?
Yes, if iCloud Photos is enabled on the iPhone. See Set up and use iCloud Photos
This process starts with an upload to iCloud of the entire collection of photos and videos on the iPhone. Thereafter, all photos taken on the iPhone are uploaded and added to this collection.
My partner is set up under family sharing but photos taken on his iPhone to not appear in the iCloud+
It sounds like his Photos app on the iPhone is not set for iCloud Photos - see above.
do I have to somehow put my Appleid on his phone to make it work?
No, that is not advisable. Each person should use their own Apple ID. You indicate that you are using Family Sharing. This allows you to share the space you rent on iCloud in your iCloud+ storage plan. See What is Family Sharing?
can one have the same Apple ID on two iPhones?
Yes, as many devices as you like - but they will all use the same account and data will be synchronised between the two iPhones - your iCloud email and iMessage texts using that account will also be set up on the other iPhone, for example. See my answer above - each person should use their own Apple ID.