You cannot navigate to a route if you have already reached it. But now that you're already there, you can just set this.tab to the value you want:

if (this.$route.name === 'AdminNotifications') {
  this.tab = 'requests';
} else {
  this.$router.push({
    name: "AdminNotifications",
    params: { tab: "requests" },
  })
}

If the component responsible for navigation is different from the component containing tab, you can push the tab parameter to $route:

if (this.$route.name === 'AdminNotifications') {
  this.$router.replace({
    params: { tab: "requests" }
  });
} else {
  this.$router.push({
    name: "AdminNotifications",
    params: { tab: "requests" },
  })
}

In the page component, replace "watcher" in mounted with an appropriate watcher that dynamically sets tab to $route.params. Any truth value of tab:

watch: {
  '$route.params.tab': {
    handler(val) {
      if (val) {
        this.tab = val;
      }
    },
    immediate: true
  }
}