How to include an emoji in my contact names?

In which app and in which menu are you trying to change your name and insert emoji characters?

The "Heart" symbol is an Emoji Character that you can add with the built-in emoji keyboard.

If you haven't already activated the Emoji keyboard, you can do it here:

Settings → General → Keyboard → Keyboards → Add New Keyboard... → search for Emoji and add.

Now you can add emoji characters to text, just like the contact on your screenshot. Look for the Emoji button in the keyboard, or tap and hold the Globe icon and select Emoji to see all the icons.

You can find more information about using emojis here:

Use emoji on your iPhone, iPad, and iPod touch - Apple Support

Arhh okay if you try to insert an emoji here:

Settings → Apple ID → Name, Phone Numbers, Email → Name

That is why. The name associated with the Apple ID must not contain emojis.

You can do this under Contacts, also under your own contact in the Contact app. But your name, which is officially associated with your Apple ID, must be without Emojis. I just tested on my own, I get the same message when I try to add an emoji to my official Apple ID name.

You can also put an Emoji on the name that people see (can choose to see) when you send them iMessages. You can do this here: Settings → Messages → Share Name and Photo

Perfect! You're welcome.