You asked two questions
- Is it possible to find from the dataset (i.e does the raw file contain information on the sample whether it is a diabetic sample or a control sample?).
- Should the diabetic and control dataset be normalized separately?
Before I continue further I'd just like to point out the graph in the Supplementary information of the Nature research article of the paper in question. This graphic has not been edited in anyway.

I will not comment further but it would be very unusual for an analyst to produce a y-axis as done here. Just to labour/labor the point this is the appendix of a Nature research article (not 'even' a Nature letter).
The answer to the first question is you can't and will need to contact the corresponding author. There is a possible work around. The reason you need to contact the authors is:
- The authors have not labeled the x-axis in any heatmap figure. This is not standard procedure and in this analysis you cannot link the heatmap values for that given mouse to the data base - because the authors have not declared which mouse is which. In addition, one cannot link the RNA-seq analysis to the proteomics data and they are paired samples, because only 3 diabetic mice are presented for the RNA-seq data but 4 mice for the proteomics data, therefore again it is not possible to link the two data sets (I understand this last point is not the precise question, but is important in the 'conclusion' below).
- The link you provided had 12 samples - not 8 as in the paper. It is very important to understand why there is a different number of samples (where one 'sample' is one mouse) in the proteomics data base compared with the apparent lower number samples presented in the paper.
To reiterate, only the authorship knows this information therefore they need to be contacted. The corresponding author is different from the data base curator who has been previously contacted (with no response). The corresponding author has to uphold the terms of publication as described at the end of the paper.
However, you still cannot link these values to the publicly available data sets, therefore until x-axis sampling is declared it is not a singular solution.
Question 2
Normalisation, I would like to point out in the appendix the authors transformed the RNA-seq data (log2) but did not do this for the proteomics data. For example aquaporins were upregulated 94 fold for proteomics data, but 4.61 fold using log2 RNA-seq data. The reason they did this is not clear and understanding this is quite important to the analysis. Why present an untransformed ratio in one analysis, but for the same experiment transform the values for precisely paired data?
The Appendix describes this information in the Supplementary Information in the following table.

Conclusion
This is a Nature paper and the authors are therefore under obligation, as per terms of publishing, to firstly supply all information on request and any information present in the published article must be explained in the Supplement Information. Those terms are stated at the end of the paper. It is clear the absence of labelling the x-axis in the heatmap in anyway (even if it is done in the legend), or the inability to link pairwise RNA-seq to proteomics data (which is very important), or the discrepancy in the samples on the database verses the samples in the paper, is not explained in Supplementary Information (Appendix).
Under these technical circumstances the authors are obliged to supply the missing information on request. You may require a chain of request escalation within your organisation because they may not respond to the initial requests, but you don't know unless you try. However, at some level in the organisation, e.g. Director, the terms of publication would be upheld by authorship in question.
First answer
But the experimental conditions are missing in the PRIDE link
The experimental conditions of the heatmap are as follows: each box is a separate animal and the heat map colors (colours) are a log2 representation of Figure A (below). In other words to understand the heat map using Figure A is the starting point. You can obtain the values from this analysis easily using a graph capture package using Figure A (not Figure B). Its a long while since I've used graph capture, so I forget the packages. If you feel there are anomalies you can then contact the authors for the precise values.
log2 is a bit weird, natural log would have been preferred:
FIGURE A
"Abundance of the indicated proteins involved in lipid synthesis, measured by proteomics, in islets isolated from control (black, Ctrl, n=4) and 2-week diabetic βV59M (white, Diab, n=4) mice. Each data point indicates a separate mouse. Mean±S.E.M. **p<0.01, ***p<0.001. HMGCS1, 3-Hydroxy-3-methylglutaryl-CoA synthase 1; HMGCR, Hydroxy-3-methylglutaryl- CoA reductase; DHCR7, 7-dehydrocholesterol reductase; ACLY, ATP citrate lyase; GPD1, glycerol phosphate dehydrogenase; ACC1, acetyl-CoA carboxylase 1; AACS, acetoacetyl- CoA synthetase; FASN, Fatty acid synthase."
FIGURE B
"Heat maps of mRNA and protein expression of the indicated lipid metabolism genes in islets isolated from control and 2-week diabetic bV59M mice. Each box corresponds to a different animal. Colour indicates log2 fold-change."
https://static-content.springer.com/esm/art%3A10.1038%2Fs41467-019-10189-x/MediaObjects/41467_2019_10189_MOESM1_ESM.pdf

Data availability
RNA data
https://www.ebi.ac.uk/ena/browser/view/PRJEB31793?show=reads
Checks out ok.
Proteomics
I checked one link only ...
http://proteomecentral.proteomexchange.org/cgi/GetDataset?ID=PXD012979
I agree something doesn't look right here just in the way it is reported, because the paper should highly polished and they did miss the link out. This could be word limit issues (Data availability might be included). The curator is listed as rom[email protected] and that would be the person to contact.