Based on a comparison of local interval correlations between the NINO34 and SOI indices, there probably is a limit to how well a model can be fit to ENSO. The lower chart displays a 4-year-windowed correlation coefficient (in RED) between the two indices (shown in upper chart):
Note that in the interval starting at 1930, the correlation is poor for about 7 years.
Next note that the ENSO model fit shows a poor correlation to the NINO34 data in nearly the same intervals (shown as dotted GREEN). This is an odd situation but potentially revealing. The fact that both the ENSO model and SOI don't match the NINO34 index over the same intervals, suggests that the model may match SOI better than it does NINO34. Yet, because of the excessive noise in SOI, this is difficult to verify.
But more fundamentally, why would NINO34 not match SOI in these particular intervals? These regions do seem to be ENSO-neutral, not close to El Nino or La Nina episodes. Some also seem to occupy regions of faster, noisy fluctuations in the index.
It could be that the ENSO lunar tidal model is revealing the true nature of the ENSO dynamics, and these noisier, neutral regions are reflecting some other behavior (such as amplitude folding) — but since they also appear to be obscured by noise, it makes it difficult to unearth.
The paper by Zajączkowska also applies a local correlation to compare the lunar tidal cycles to plant growth dynamics. There's a treasure trove of recent research on this topic.