If you're willing to constrain the statistical question a little: Instead of asking whether "the rank ordering between the two groups is significantly different," you could ask the following four questions.
Is there an effect of group on the ranked value of Animal 1?
Is there an effect of group on the ranked value of Animal 2?
Is there an effect of group on the ranked value of Animal 3?
Is there an effect of group on the ranked value of Animal 4?
Each of these can be answered by a Mann-Whitney U test (under 'T tests' > 'Independent Samples T Tests,' in jamovi). There resulting p values can then be adjusted for multiple-comparisons (e.g., using p.adjust function in R).