edited: Dan beat me to it, to to add to his remarks;
A geoprocessing function such as split at point will always create an output feature class like that. My suggestion is to add an attribute of Long Int type and call it something like Key to your original data. Calculate Key = OBJECTID. That gives them all a unique value that will persist after you split.
After the split, add all the fields you want from your original feature class to your new feature class. You now you have the basis with which to perform a Many to One join: Your new feature class will have many instances of to original Key value, while the original feature class has only one instance of the Key value.
With the join in place:
Calculate NewFeatureField_A = OldFeatureField_A
Calculate NewFeatureField_B = OldFeatureField_B
Calculate NewFeatureField_C = OldFeatureField_C
etc, etc.
And remove the join when you are done.
You can add your own ID field if you like. See a post of mine that discusses GUIDs: Calculate a GUID Field with a value
That should just about do it....