In Settings > Messages tap Send& Receive, then your Apple ID. Sign out of Messages. Do this on on both the iPhone and iPad. Wait a few minutes and sign back in on both devices. You should receive notifications (not to long after) confirming that that iPhone and iPad are now set to receive messages from your Apple ID and iPhone number on both devices, and possibly other email addresses. You can double check this by going to the same place and seeing what email addresses and phone number appear.
I've added some extra info below, incase you're not up on the differences between the iPad and iPhone and how Messages will behave.
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The Messages app on the iPad is only capable of using Apple's iMessage service. This does not use the iPad's phone number. It uses your Apple ID/email address and only works between Apple devices.
On an iPhone you the Messages app uses both iMessages and SMS, switching between them when needed (if enabled).
When you send a text on an iPhone your sent text shows up as blue if you are using iMessages and green if you are using SMS. The iPad never sends SMS, and so your text is always blue.
SMS is the texting service provided through your cell phone service. It works between any cell phone, provided both users have SMS enabled on their phones. The iPad does not have this service. Despite having a phone number, that number is not activated as an actual phone, therefor it does not give access to cell phone services, including SMS (or MMS).
To add further confusion:
If you have both an iPhone and an iPad registered to the same Apple ID, an iMessage (Apple Service) text sent to your iPhone's number can show up (if enabled) on the iPad, but, SMS messages sent to your iPhone will not show up on the iPad. Ever. This is because SMS goes directly to your phone, bypassing Apple's iMessage servers.
There are third party apps you can use to send and receive SMS messages on your iPad. Some are free and some are paid, but they all have one thing in common. They give you phone number.
