Short answer: yes they do but there are ways of possibly telling them apart by seeing epigenetic differences. Meaning that in some cases it’s possible to see that factors due to lifestyle differences can in fact change how some genes are expressed. It’s safe to say though that yeah they will have a pretty near identical genome (because their sequence will be so similar) but seeing things like non-coding RNA, Chromatin Structure Assays, Histone Modification, and DNA methylation can sometimes help on differentiating monozygotic twins. TLDR: their genome is identical but looking at the DNA structure/expression could possibly tell them apart in some cases.
