You can create new customers using the customers admin REST API: https://help.shopify.com/en/api/reference/customers/customer
Or you can use the storefront API: https://help.shopify.com/en/api/custom-storefronts/storefront-api/reference/mutation/customercreate.
Logging in however cannot be done with any APIs besides multipass, which is for Plus shops only.
Does the above not suit your needs? Take a look at the documentation if you haven't already and let me know if there's still something the above does not address.
If you're creating a mobile app, have you looked at either of the mobile buy SDKs for Android and iOS? They both support customer logging in via the customerAccessTokenCreate mutation (which is available with the storefront API if you don't want to use the SDKs):
I hope that helps you out a bit.
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