Like most API that requires API key such as Google map, you shouldn't let users to be able to communicate directly with them.
You need to have your own server and make each user create username and password. Users would have to log in to your server in order to use the Firebase feature that requires payment. When they log in to your server, your server is responsible for making the requests and sending the request result back to Unity application. Unity app makes request to your server, your server checks if limit has been reached on this account then decides to make the actual request and returns the result.
With your server making the request, you will be able to implement limit on each account. By doing this, you will also be able to place limit on IP Addresses and make sure that new accounts are not created or used from the-same IP to circumvent the server limit when the limit has been reached.
EDIT:
The answer above still apply to many other API that requires API keys. Luckily Firebase has a feature called "Firebase Security Rules" and this might be useful in your case. You can use "Firebase Security Rules" to create a rate-limiting rule that each time there is a request, you can check the timestamp and then update the database with the current time of the request. With this you can implement something like 1 requests per minute.
There is a simple example here. There is also more complicated and complete example here too.