How can I merge my iPhone with my iPad using a different Apple ID

Merging Apple accounts (IDs) or transferring data is not officially supported, nor can it be done easily or completely. In your situation it is possible to keep and merge some data by selecting options to keep data when signing into a device with a different ID.

Apr. 2018 post by ASC helper Winston Churchill on transferring data from one Apple ID to another. https://discussions.apple.com/message/33333380#message33333380

Feb. 2021 tips from Eric Root https://discussions.apple.com/thread/252468781?answerId=254653802022#254653802022

Look into using Family Sharing to continue to be able to use items purchased with one Apple ID with other Apple IDs.

Family Sharing - https://support.apple.com/HT201060

Set up Family Sharing - https://support.apple.com/HT201088

Fortunately when you sign out of one ID and sign in with a different one it asks you in most cases if you wish to retain the data from it on your device. Say yes and then delete individual items you do not want.

Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac --> https://support.apple.com/HT208242

If another Apple ID is signed in you may need to enter the password for that ID to sign out of that ID first.

Click on this link to learn how to 'Sign in with your Apple ID' --> https://support.apple.com/HT204053

If you are talking about Apple purchases (which you do not say) then those cannot be transferred, but there are many things that can otherwise be associated with an Apple ID that can be transferred. Apple's recommendation just goes with the extremely narrow interpretation of your question. Reality is something else.

I transfer stuff all the time from one Apple ID to another. For example, photos by using AirDrop. I don't even need Family Sharing which does let you transfer/share a lot more, including even purchases.

Content from one Apple ID cannot be merged or transferred to another Apple ID.