There were actually multiple scientific studies done on the fish ingot which led to the design being changed to maximize dietary iron. However, some of the studies were published in predatory journals and the peer review on these is dubious. Here are the more reliable publications:
Iron-deficiency anaemia in rural Cambodia: community trial of a novel iron supplementation technique
Blood iron levels were higher in women in the iron fish plus follow-up at 3 months compared with controls, but this was not maintained. At 6 months, haemoglobin and serum iron had fallen in all groups and the proportion of anaemic women had increased.
I assume this study was done with an early prototype, as the follow-up has better results:
Happy Fish: A Novel Supplementation Technique to Prevent Iron Deficiency Anemia in Women in Rural Cambodia (PhD thesis)
Significant improvements in serum ferritin concentration were observed at 9 months (6.9 ng/mL) and endline (30.8 ng/mL) in women who used an iron fish regularly when compared to the control group. Overall, use of the iron fish led to a two-fold reduction in the prevalence of anemia
Review of Iron Supplementation and Fortification (journal of questionable quality; possible conflict of interest from one author)
While this has potential to be effective in addressing iron deficiency it has only been tested on
women of reproductive age, and thus it is not certain that the Lucky Iron Fish will provide the
right amount of iron for children. The clinical trial included only 6 pregnant women, and future
research could seek out larger numbers of pregnant and lactating women to test the effica‐
ciousness in these times of high iron demand. Furthermore, gaining a better understanding of
the type of iron that leaches into water and foods would be beneficial, as would further testing
of fortifiable foods. Additional challenges associated with this approach are predominantly
surrounding acceptability and education, as this intervention requires significant behavior
change in home practices. Research regarding the acceptability, adherence and cost-effective‐
ness of this strategy should take place in order to carefully compare it with alternate inter‐
ventions to be included in a national nutrition strategy.