Ory is the largest open source community in the world for cloud software application security. We maintain advanced open source security software solving authentication, authorization, access control, application network security, and delegation. Ory implements a variety of industry and best-practice standards including OAuth 2.0 / OAuth 2.1, OpenID Connect, Zero Trust Networking, Google Zanzibar Policy Framework, FIDO2 U2F, WebAuthn, TOTP, and more.

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Is there any nextjs 13 custom login implementation which uses the app router?

No, we currently only have an example that uses the page router.  <https://github.com/ory/kratos-selfservice-ui-react-nextjs>

Contributions for an app-router example are welcome, though!