Does the SIM card have to switched?

Carriers with support for “quick start” will automatically migrate the telephone number without swapping the SIM, this happening as part of the data-migration process.

Details:

Transfer an eSIM from your previous iPhone to your new iPhone - Apple Support

Check with your carrier.

If you need to, swap the SIM, if the SIM sizes match:

Learn which size SIM card your iPhone or iPad uses - Apple Support

If the SIM sizes don’t match and you can’t switch to and use the eSIM, you’ll need either an Apple SIM, or a SIM from the carrier.

And as mentioned above, the target iPhone must either not be carrier-locked, or the SIM must match the carrier lock:

How to unlock your iPhone for use with a different carrier - Apple Support

Swapping the SIM is the quickest and simplest way to get a new device onto your account with your cellular service provider. Especially if you’re buying a device from Apple and paying in full for an unlocked device. In that case, you’re carrier has no knowledge of your purchase and change of device and swapping the SIM will immediately update your account and enable it on that device for your existing cellular service account.

If you already have a nano SIM card, you can use the SIM in the new iPhone.

Assuming the new iPhone is not carrier locked to another carrier.