For content that’s more interesting to you, you can pin it in the Messages app. This will help you keep track of it in Shared with You, Messages search, and the Details view of the conversation.
How to Pin Content in Messages
Open a conversation in Messages
Tap and hold the particular piece of content you want to pin
Tap the Pin button
How to find the pinned content:
Go to the same conversation in Messages where the content was pinned
Tap the name at the top of the message thread
Scroll down until you find the Pins section
Alternatively, you can tap the Search bar in Messages to see the Pins section.