Prokaryotes are hugely diverse in base composition (ratio of A + T to G + C for example) and eukaryotes are less diverse in that regard. But there is certainly more diversity between different prokaryotes than the difference between eukaryotes and prokaryotes. Also, the eukaryotes have nuclear genes with introns and all, and also plastid (chloroplast and mitochondria) genes that are derived from prokaryotes and don’t have introns. The eukaryote nuclear genes are more similar to some of the archeabacteria than they are to the eubacteria which has lead to the hypotheses (now pretty well proven) that eukaryotes are a fusion of archaebacteria plus eubacteria.
You can’t take any random tiny bit of eukaryote or prokaryote DNA and determine where it came from just by base composition or codon use alone. However, using BLAST to compare your sequence against all sequences in GenBank you may indeed be able to tell exactly which organism it came from. The database now has the complete genomes of many thousands of different organisms.