Which service should be recommended for an app that requires external identity integration with customizable branding?

The recommended service for an app that requires external identity integration with customizable branding is Azure Active Directory B2C. This service is specifically designed for scenarios where businesses need to manage customer identities and engage with customers using various identity providers.

Azure AD B2C offers the ability to create branded user experiences for sign-up, sign-in, and profile management. Organizations can customize the user interface to reflect their brand identity, providing a seamless experience for users. Additionally, it supports a range of identity providers such as social accounts (e.g., Facebook, Google) and local accounts, allowing extensive flexibility in managing user identities.

In contrast, while Azure AD Connect is geared towards synchronizing on-premises directories with Azure Active Directory, it does not focus on external customers or branding. Azure Logic Apps and Azure Functions serve different purposes related to workflow automation and serverless computing, respectively, and are not focused on identity management or branding capabilities. Therefore, for customizable branding and external identity integration specifically, Azure Active Directory B2C is the most appropriate choice.