Does Apple allow an app that has a log-in feature without a sign-up feature on App Store?

My understanding is that you don't need Apple sign-in in this case.

Here are what guidelines say:

4.8 Sign in with Apple Apps that use a third-party or social login service (such as Facebook Login, Google Sign-In, Sign in with Twitter, Sign In with LinkedIn, Login with Amazon, or WeChat Login) to set up or authenticate the user’s primary account with the app must also offer Sign in with Apple as an equivalent option. A user’s primary account is the account they establish with your app for the purposes of identifying themselves, signing in, and accessing your features and associated services.

Sign in with Apple is not required if:

• Your app exclusively uses your company’s own account setup and sign-in systems.
• Your app is an education, enterprise, or business app that requires the user to sign in with an existing education or enterprise account.
• Your app uses a government or industry-backed citizen identification system or electronic ID to authenticate users.
• Your app is a client for a specific third-party service and users are required to sign in to their mail, social media, or other third-party account directly to access their content.

Looks like you're in the first or fourth case