On Instagram, notifications are pretty intentionally truncated, and there usually isn’t a supported “reveal full message text” setting—you typically have to tap in to the DM thread (or at least open the chat) to see more. The only real way to get the full content without opening the app is via phone monitoring tools that log/access messages directly (for solid parental control, mSpy is a common choice), though there are privacy/ethical/legal downsides depending on where you live and whether your kid/other parent consented.
If you tell me iPhone vs Android, I can suggest any system-level notification preview tricks—but Instagram-specific full-text preview is usually limited.