What is the structure of my Linear mixed effect model? I'm planning to use a LMM for my analysis. This is new to me.
I wonder what structure my LMM has? Is it a nested random factor LMM?

As I tried to show in the graphic already, I took body length and isotopic values from organs of every individual in my sample collection.
And how can I classify it according to the McGraw and Wong (1996) Convention? In order to choose the correct ICC?

Ecological data from what I understand is often non-normally distributed, so it may be best to employ a generalized linear mixed model (GLMM) instead. Your model is almost correct, but you need to nest the individual fishies in their respective species. Without any other specification of the model, it would minimum look something like this:
glmer(isotopic_value ~ organ + body_weight_g + (1|species/id)

This includes random intercepts only. For more complicated random effects modeling, this paper written by the author of lme4 has all the necessary syntax on Page 7. A useful paper on GLMMs in biology and ecological data can be found here as well.
I'm not sure what you mean by the McGraw and Wong convention, but feel free to add a comment here and I can explain if I have an appropriate answer. ICC can be found easily for GLMMs with the performance package, using a code like this:
icc(model)

Edit
Dipetkov made two really good points that I hadn't considered when I looked through your question. First, you only have two species here, which is much better modeled as a fixed effect (typically you want more than 5 clusters for random effects and you only have 2).
He also mentioned that because of the way I have modeled the HLM, it assumes your other IVs are similar among species. This was what I alluded to when I mentioned using the syntax from the doc I shared for more specific random effects. For example, perhaps you believe that species have varying slopes for body weight in grams. You could alternatively model it like so:
glmer(isotopic_value ~ organ + body_weight_g + species + (1+body_weight_g|id)

You will need good theory-driven reasons for your HLM design, which will drive your random slope/intercept terms.
