Fresh Laravel Sanctum API redirecting to '/dashboard' route

You are right, AppProvidersRouteServiceProviderRouteServiceProvider::HOME const is set to "/dashboard". If you follow the usage of this const you can see that is used by AppHttpMiddlewareRedirectIfAuthenticated middleware

class RedirectIfAuthenticated {
/**
 * Handle an incoming request.
 *
 * @param  Closure(IlluminateHttpRequest): (SymfonyComponentHttpFoundationResponse)  $next
 */
public function handle(Request $request, Closure $next, string ...$guards): Response
{
    $guards = empty($guards) ? [null] : $guards;

    foreach ($guards as $guard) {
        if (Auth::guard($guard)->check()) {
            return redirect(RouteServiceProvider::HOME);
        }
    }

    return $next($request);
}
}

This middleware is aliased as "guest" in the AppHttpKernel:

protected $middlewareAliases = [
    'auth' => AppHttpMiddlewareAuthenticate::class,
    'auth.basic' => IlluminateAuthMiddlewareAuthenticateWithBasicAuth::class,
    'auth.session' => IlluminateSessionMiddlewareAuthenticateSession::class,
    'cache.headers' => IlluminateHttpMiddlewareSetCacheHeaders::class,
    'can' => IlluminateAuthMiddlewareAuthorize::class,
    'guest' => AppHttpMiddlewareRedirectIfAuthenticated::class,
    'password.confirm' => IlluminateAuthMiddlewareRequirePassword::class,
    'signed' => AppHttpMiddlewareValidateSignature::class,
    'throttle' => IlluminateRoutingMiddlewareThrottleRequests::class,
    'verified' => AppHttpMiddlewareEnsureEmailIsVerified::class,
];

And this middleware is used in unauthenticated routes defined by routes/auth.php

Route::post('/register', [RegisteredUserController::class, 'store'])
            ->middleware('guest')
            ->name('register');

Route::post('/login', [AuthenticatedSessionController::class, 'store'])
            ->middleware('guest')
            ->name('login');

Route::post('/forgot-password', [PasswordResetLinkController::class, 'store'])
            ->middleware('guest')
            ->name('password.email');

Route::post('/reset-password', [NewPasswordController::class, 'store'])
            ->middleware('guest')
            ->name('password.store');

This built-in middleware middleware will try and redirect you to the HOME if you are already authenticated.
I think that there is an issue in Laravel breeze scaffolding (php artisan breeze:install api) that not provide a check on the response type in the
AppHttpMiddlewareRedirectIfAuthenticated middleware.
If you see instead the AppHttpMiddlewareAuthenticate middleware here is automatically provided a redirectTo method that check for the expected response ($request->expectsJson()):

class Authenticate extends Middleware
{
    /**
     * Get the path the user should be redirected to when they are not authenticated.
     */
    protected function redirectTo(Request $request): ?string
    {
        return $request->expectsJson() ? null : route('login');
    }
}

TLDR
In short to work correctly with the AppHttpMiddlewareRedirectIfAuthenticated middleware in API based authentication you can add manually the following check in the middleware:

foreach ($guards as $guard) {
if (Auth::guard($guard)->check()) {
  if ($request->expectsJson()) {
    return response()->json(['error' => 'Already authenticated.'], 200);
  }
  return redirect(RouteServiceProvider::HOME);
}
}