Welcome Jazzamine,
Thank you for using Apple Support Communities. Based on your post, you can't download or save photos that have been shared with you through the Messages app. We'll do our best to assist with this.
All of these photos should appear when you select on the contact name at the top of the conversation and scroll down. You should be able to save them from within this page. However, it sounds like you aren't seeing all of the content shared.
With that in mind, try opening up the Photos app > 'For You' tab at the bottom. Then scroll down to 'Shared with You'. This folder should have content shared with you from the Messages app. Do you see all of these photos there? See more here: Use Shared with You on your iPhone, iPad, and iPod touch
If not, when you go to Settings > Messages, how long are you saving messages? This should be under 'Message History' and 'Keep Messages'. If it's selected on a specific timeframe like 30 days or 1 year, could they have been removed due to this setting?
Hoping to hear back, take care.