In my impression, initially many people did not believe that an Uranium nucleus disintegrates into two halves when bombared with neutrons.
Experimentally, there must be some difficulty to determine the products of the nuclear reaction. What precisely are they? Is it because the products are too little, i.e., with atom number orders of magnitude smaller than $10^{23}$?
Because fission products are smaller atoms than the fissile atom, they are chemically distinct and therefore easily isolated and identified. Identifying the "daughter elements" was not the limiting factor in understanding nuclear fission processes in the early days of the field.
If you read about the discovery of Nuclear Fission$^{[1]}$, It says that their initial hypothesis after discovering neutron was to make elements bigger than Uranium by bombarding it with neutrons.
Since neutron is not big compared to Uranium's nucleus it shouldn't cause much change. It was later on that they realised that they had broken the Uranium atom into smaller atoms. From there I suppose the discovery and study of nuclear fission wasn't far since they already had knowledge of alpha and beta decay.
1 https://www.aps.org/publications/apsnews/200712/physicshistory.cfm
It depends what you mean by "fission product".
A product fission is usually defined as an isotope produced by the fission of a nuclide after prompt deexcitation. If we talk about the isotope that are produced at the moment of the fission, the term is "fission fragment". The fragment are mainly produced in an excited state that will deexcite by emitting gamma-rays and/or neutrons.
Historically, the fission products have been identified using chemical analysis. Yet, chemical analyses give only access to the chemical nature of the isotope, i.e. its number of protons.
To determine the number of neutrons of a fission product, physicists now use gamma spectrometry.
Yet, gamma spectrometry suffers some limitations; for example, one cannot detect fission product that become stable faster than the gamma measurement.
To solve this issue, it is possible to use gamma spectrometry with mass spectrometry and thus physicists access to more isotope but not all.
Nowadays, physicist use mass spectrometer on particle accelerator in inverse kinematics to measure the number of protons, the number of neutrons (see this article or this one for some examples). Physicists may also use gamma spectrometers located around the collision area to measure the excitation energy of the fission fragment, i.e. before any neutron evaporation or gamma emission. From these raw measurement, decay calculation can be used to extract fission product properties after a given duration.
