Could you please clarify if these imperial constitutions are the same as the Constitutions adopted by different States throughout the contemporary world?

No, and it's not really correct to say that the imperial Codices of the late Empire preserve earlier constitutions, since strictly speaking there were no such things. We refer to such written documents as constitutions, but they were not like, e.g. the U.S. Constitution in that they did not consist of a written document intended to lay out the guiding principles of a governing system. The ancient constitutions established the form of government, but only in so far as that they cataloged the laws of the state. That's what we mean when we say "constitution" in antiquity, we mean the sum total of a state's laws. The late antique codices were the first written version of this catalog, as earlier the ancient world had known no such thing as a written constitution--instead, individual laws were written but never compiled into a single official text. Now, mind you, Justinian's text does not in fact preserve all the laws, nor does it preserve their wording. Instead the corpus iuris (including the Codex, Digest, and Institutes) was a collection of the writings of earlier jurists, authors who compiled what are essentially case studies of various laws and sought to explain their practical action. So Justinian preserves a number of very archaic laws as well as quite a lot of conflicting views on laws. Moreover, the corpus iuris contains precious little about actual government institutions, which is what we expect from a modern constitution. Rather the bulk of it (and the entirety of the Digests) lists civil law, and a little criminal and administrative law. Nowhere in the corpus iuris does it say how the state's powers should work or how various government institutions should be run or interact with each other for the purposes of governing. And it is not a collection of ideals for the government, but a collection of actual laws that were actually in effect at the time of the document's completion. In that it doesn't really resemble a modern constitution at all, except in that technically speaking it's the most basic legal document (but it crucially does not establish any foundational governing principles, which is the purpose of a modern constitution)