fix(docs-infra): fix deploy-to-firebase.sh for master and v10.0.x branches (#37762)

The deployment to aio is currently failing because #37721 introduced
"project" entry into the firebase.json which means that we now need to
select the deployment target before deploying to firebase.

This change fixes the issue and refactors the file to be easier to read.

I also added extra echo statements so that the CI logs are easier to
read in case we need to troubleshoot future issues.

PR Close #37762

aio/scripts/deploy-to-firebase.test.sh

@@ -68,6 +68,7 @@ function check {
   expected="Git branch        : master
 Build/deploy mode : next
 Firebase project  : aio-staging
+Firebase site     : aio-staging
 Deployment URL    : https://next.angular.io/"
   check "$actual" "$expected"
 )
@@ -103,6 +104,7 @@ Deployment URL    : https://next.angular.io/"
   expected="Git branch        : 4.3.x
 Build/deploy mode : stable
 Firebase project  : angular-io
+Firebase site     : angular-io
 Deployment URL    : https://angular.io/"
   check "$actual" "$expected"
 )
@@ -139,6 +141,7 @@ Deployment URL    : https://angular.io/"
   expected="Git branch        : 2.4.x
 Build/deploy mode : archive
 Firebase project  : v2-angular-io
+Firebase site     : v2-angular-io
 Deployment URL    : https://v2.angular.io/"
   check "$actual" "$expected"
 )
@@ -158,7 +161,8 @@ Deployment URL    : https://v2.angular.io/"
   )
   expected="Git branch        : 9.1.x
 Build/deploy mode : archive
-Firebase project  : v9-angular-io
+Firebase project  : aio-staging
+Firebase site     : v9-angular-io
 Deployment URL    : https://v9.angular.io/"
   # TODO: This test incorrectly expects the Firebase project to be v9-angular-io.
   #       v9-angular-io is a "multisites" project currently within the aio-staging project