All of these have a monthly storage cost, and so I will only mention the read/write costs for each product.
Amazon Aurora Serverless is a SQL database that is priced based on the amount of "Aurora capacity units" you use for reads and writes with per-second billing. The database “pauses” itself while not in use, and you don’t pay for ACUs while the DB is paused. There is a minimum billed amount of 5 minutes every time it starts up.
Amazon Timestream (currently only in preview) is a NoSQL database that is specifically optimized for storing time-series data. Pricing is based on the size of the data that is written or read during a request.
Strictly speaking, Amazon S3 is not a database—it's an object store—but it has a per-request pricing model. You can use Amazon EMR (per-second billing, one minute minimum) or Amazon Athena (per request billing, based on the amount of data scanned) to query/analyze data that is stored in S3.
Azure CosmosDB is a NoSQL database and has a per-request pricing model. it turns out that you have to pay for hourly provisioning of the request units.
GCP Big Query is an Analytics Data Warehouse. It has two pricing models, including a "pay for only what you use" pricing model that based on the amount of data read or written.