As a matter of law, the Duchy is correct, because a superior court of record agreed, and dismissed the arguments made on the other side.
The cited claims from Republic come from a case in the First-Tier Tribunal (EA/2010/0182) about whether the Duchy was a "public authority" for the purpose of the Environmental Information Regulations 2004. The FTT said that it was, since either the Duchy or the Duke were carrying out certain functions of public administration (as the harbour authority for St Mary's Harbour on the Isles of Scilly), and whichever one it was, the Duke was in charge of the Duchy; and meanwhile, the Duchy appeared to have some elements of legal personality, as cited in the question above. This decision was appealed to the Upper Tribunal, The A-G for the Prince of Wales v the IC and Mr Michael Bruton  UKUT 0154 (AAC), which disagreed as to the Duchy being a body with its own separate identity - see paragraphs 71-102 - and reversed the ruling below.
The basic issue is that the Duchy of Cornwall was brought into being by a royal charter of 1337. It predates modern common-law or statutory developments about bodies corporate, income tax, corporation tax, and so on. It also precedes the not-exactly-modern developments of the limited scope of royal authority and the prerogative, norms of statutory construction, the Crown as separate from the royal person, and other apparatus of constitutional monarchy. It is true that it sits uncomfortably in the modern world, as a sui generis feudal creation that is not something we would create today. However, as a matter of law, the UT said that what counts is what the Duchy is, not what it ought to be. Figuring out the law means looking at the actual law as opposed to ancillary indications, such as what Duchy officers have said from time to time, or administrative actions of the Health and Safety Executive. Parliament has had the opportunity to consider the position of the Duchy several times and has so far declined to change it.
The charter of 1337, which was deemed to be an Act of Parliament by a court case in 1606 (!), grants land and privileges to the Duke of Cornwall, whoever that might be from time to time, and with a reverter to the Crown when there is no Duke. The UT found that this "creates something that has similarities to a trust, but without trustees" (there was a nascent body of law relating to trusts in 1337, but it was nowhere near its modern flourishing). Unlike with the creation of the Duchy of Lancaster, this charter does not incorporate the Duchy as an entity in its own right - the "Duchy" is just a word used to describe the possessions of the Duke or King held under the charter. If it is a separate entity then that is not because of the charter.
The UT examined various of the claims made in the FTT but rejected them, saying that "such examples cannot create such an entity or body and each investigation of the submissions based on them that took place before me showed such statements or implications to be wrong (e.g. that the Duchy owns Highgrove was shown to be wrong by the
production of the Land Certificate)." It also rejected the idea that the people who administer the Duchy could be regarded as some form of partnership or unincorporated association, finding that they had not come together in that fashion. It said that therefore, the only remaining route to the Duchy being a corporation would be if the other primary legislation dealing with its management had created it as such by "necessary implication".
Those Acts, of 1844, 1863 and 1982, deal with accounting and management for the Duchy of Cornwall. From their context and content, the UT said that they seem to envisage the property in question as belonging to the Duke (not the Duchy) as "the legal owner of the assets etc. comprising the Duchy estate", and that "the relevant officers and employees work for the Duke, or the Duke’s guardian when he is a minor, or the Sovereign when there is no Duke, and not for an entity or corporation called the Duchy." Therefore:
[T]he Duchy of Cornwall or the Duchy is no more than a name that has been used correctly to describe the possessions of the Duke of Cornwall (the Duchy estate), or to the Duke (or his title), or collectively and conveniently to describe the officers and persons who from time to time act for and on behalf of the Duke as the owner of the Duchy estate.
Save to the extent that it is a reference to the Duke as a natural person the Duchy of Cornwall has no legal personality and is not a person, body or entity that has its own functions and powers or a separate identity of its own.
Although this was not before the UT, it would seem to follow for tax purposes that if the Duchy is not a body corporate, then it is not liable for corporation tax under the Corporation Tax Act 2010 - see section 1121, which defines "company" for that and the companion Corporation Tax Act 2009. It also does not fall within section 443, "Companies controlled by or on behalf of Crown", firstly because if it's not a company, then that section fails immediately, and secondly because "Crown" here refers to the Crown in the sense of governmental public functions, not the Queen (or her family members) personally.