Hi Alejandro,
Your question: How do I know which date each shot corresponds to? is actually more complex to answer than you may initially think with the way the APIs work with requests that are using a time range. This is because it is possible that multiple acquisitions could be used in the returned image from Process API. I’ve put together an example for you to test this out:
curl -X POST https://services.sentinel-hub.com/api/v1/process -H 'Content-Type: application/json' -H 'Authorization: Bearer <access_token>' -d '{ "input": { "bounds": { "bbox": [ 13.388144, 41.23186, 13.485986, 41.298698 ] }, "data": [ { "dataFilter": { "timeRange": { "from": "2023-09-22T00:00:00Z", "to": "2023-09-24T23:59:59Z" }, "mosaickingOrder": "leastCC" }, "type": "sentinel-2-l2a" } ] }, "output": { "width": 631.466, "height": 573.591, "responses": [ { "identifier": "default", "format": { "type": "image/jpeg" } } ] }, "evalscript": "//VERSION=3\n\nfunction setup() {\n return {\n input: [\"B02\", \"B03\", \"B04\"],\n output: { bands: 3 }\n };\n}\n\nfunction evaluatePixel(sample) {\n return [2.5 * sample.B04, 2.5 * sample.B03, 2.5 * sample.B02];\n}" }'
This returns:
You can see that this actually returns data from two dates even though the time range is limited to three days. If it is important to you to know the date of the image that you return, I suggest you limit the time range to a single date in your request.
To obtain the dates, you can use Catalog API to obtain these dates. An example of how to make requests for several timestamps using Python can be found here. I think that should help you with what you are trying to achieve 
If you are still stuck, then please reach out to us!