feat(auth): 앱(네이티브) 구글 로그인 — @capgo/capacitor-social-login

- 웹은 기존 GIS 유지, Capacitor 네이티브(APK)는 네이티브 구글 로그인 사용
  (웹뷰에서 GIS 팝업/FedCM 미동작 대응)
- 받은 ID 토큰은 기존과 동일하게 백엔드 /api/auth/google 로 검증
- src/lib/googleAuth.ts + LoginPage 네이티브/웹 분기

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

src/pages/LoginPage.vue

@@ -12,8 +12,20 @@
     </div>
 
     <q-card flat class="login-card q-pa-lg">
-      <!-- 구글 로그인 버튼(GIS 렌더링 영역) -->
-      <div ref="googleBtn" class="flex flex-center q-mb-md"></div>
+      <!-- 구글 로그인: 앱(네이티브)은 버튼, 웹은 GIS 렌더링 -->
+      <q-btn
+        v-if="isNative"
+        class="full-width q-py-sm q-mb-md"
+        color="white"
+        text-color="grey-9"
+        unelevated
+        icon="mdi-google"
+        label="Google로 계속하기"
+        :loading="googleLoading"
+        :disable="!googleClientId"
+        @click="onGoogleNative"
+      />
+      <div v-else ref="googleBtn" class="flex flex-center q-mb-md"></div>
       <div v-if="googleError" class="text-caption text-negative text-center q-mb-md">
         {{ googleError }}
       </div>
@@ -63,6 +75,7 @@ import { useAuthStore } from '@/stores/auth'
 import { authApi } from '@/api/auth'
 import { ApiError } from '@/api/http'
 import { isAppleSignInAvailable, signInWithApple } from '@/lib/appleAuth'
+import { isNativeGoogleAvailable, signInWithGoogle } from '@/lib/googleAuth'
 
 const router = useRouter()
 const route = useRoute()
@@ -71,6 +84,9 @@ const auth = useAuthStore()
 
 const googleBtn = ref<HTMLElement | null>(null)
 const googleError = ref('')
+const googleClientId = ref('')
+const googleLoading = ref(false)
+const isNative = isNativeGoogleAvailable()
 const appleLoading = ref(false)
 const appleAvailable = isAppleSignInAvailable()
 const devLoading = ref(false)
@@ -85,6 +101,9 @@ async function initGoogle() {
       googleError.value = '구글 로그인이 아직 설정되지 않았습니다.'
       return
     }
+    googleClientId.value = clientId
+    // 앱(네이티브)은 버튼 클릭 시 네이티브 로그인 → GIS 초기화 불필요
+    if (isNative) return
     await loadGsi()
     const google = (window as unknown as { google: any }).google
     google.accounts.id.initialize({
@@ -124,6 +143,21 @@ function loadGsi(): Promise<void> {
   })
 }
 
+async function onGoogleNative() {
+  if (!googleClientId.value) return
+  googleLoading.value = true
+  try {
+    const idToken = await signInWithGoogle(googleClientId.value)
+    await auth.loginGoogle(idToken)
+    redirectAfterLogin()
+  } catch (e) {
+    if (isAppleCancel(e)) return
+    notifyError(e)
+  } finally {
+    googleLoading.value = false
+  }
+}
+
 async function onApple() {
   appleLoading.value = true
   try {