Can experiments in ELISA kits be monitored via both fluorometry and photometry?
can we use both methods depending on whether the marker enzyme on the antibody is given a fluorescent or chromogenic substrate?


This is essentially correct. 

You can have the same ELISA work on different methods of detection. The most common ELISA is colorimetric, and works by binding a secondary antibody conjugated to some enzyme to your primary antibody. Fluorescent ELISA platforms are less-widespread but only because the colorimetric ELISA is reputed and "gold standard."

Horseradish peroxidase (hrp) is a common conjugate system utilized in sandwich ELISAs, and by adding TMB solution to the reaction wells, the resulting reaction produces a diimine that turns the liquid in the well blue, and does so in proportion to the amount of analyte captured. You stop the reaction by adding an acid, which turns the well yellow, and you read the OD at 450nm (they also recommend you subtract the OD at 510nm to account for the light scattering of the microplate).



Figure 1. R&amp;D Systems Quantikine assays, HRP-streptavidin detection. https://www.rndsystems.com/products/quantikine-hs-colorimetric-sandwich-elisas-high-sensitivity

On the other hand, we know that we can also conjugate the antibody with a fluorescent molecule if we wanted to, like FITC or PE. All we need to do is excite the molecule with an appropriate light source or laser to get a fluorescent emission, and read with an appropriate system to read those signals. The fluorescent intensity would then be proportional to the captured analyte, based on a standard curve just like any ELISA.

So the real answer is we can use one, the other, or both, just not at the same time. 

Just be cognizant that different platforms can yield different results, so you have to validate your two platforms, and correlate the two to ensure reproducibility. Most labs just try to stick with one or the other, though.