In the US, only ‘on’ is correct in the given sentence.
You could possibly replace ‘on the same day’ with something like ‘within the span of a single day’ but it doesn’t add anything.
You could also use ‘in’ with a different meaning of ‘day’ where you are referring to ‘daytime’. ‘Daytime’ and ‘nighttime’ are the portions of a day when the sun is up and down, respectively. For example, ‘In (the) daytime, temperatures are pleasant, but be prepared for cold nights.’ (Note that in my example, it is possible to omit the ‘the’ before daytime. Doing so gives the sentence a slightly more general sense.)
Replacing ‘daytime’ with ‘day’ in my example is not a good or precise way of speaking, but it is something that you could probably hear native speakers say. From the context, this is not the meaning of ‘day’ in your example, so only ‘on’ works in your example.