I still have 'em on the old phone (Icon) and want to put them on the A20S. Supposedly there's something for that under settings but I couldn't find it. Maybe it has something to do with no sim. I use it wifi only.
If you don't want to sync them then the simplest way is to go into the contacts app and look in its settings for an "export" option (exactly where it will be will depend on the particular app, "contact management" settings is a common place, often appears as "import/export"). This will let you export your contact book as a vcf file. Copy that to the new phone and then go into the new phone's Contacts app's settings and "import" them.
The key thing is that it's in the contacts app's settings, not the phone's settings.
"vcf" is the file extension for "vcard file". Vcard is the standard contact storage format, so you can read this into any contact manager. It will probably just write it to the top directory of your internal storage, though conceivably a different manufacturer's app could put it in a different place.
Do you mean why not in the Phone app's settings or the device's system settings? If you mean the phone app, that's because the contacts app is the thing that manages contacts. The Phone app is just one of the apps that uses them (message apps, email apps etc also use them), so the Contacts app is the logical one to do this. If you mean the system settings, they don't generally cover things that affect specific apps (though Google make exception for themselves in places). Putting contact management there would be no different from putting email settings, calendar display options, and once you start that your system Settings just becomes huge.