No legal requirement for a studbook passport
Although having an ID passport for an equine that is in the EU is a legal requirement, having the equine's breed and pedigree recorded in the passport is not and is therefore optional and it is therefore the owner's right to decide where (or even if) they want the equine's breed and pedigree recorded. It is therefore implicit in EU equine identification legislation that once an equine in the EU of any breed has been issued a basic ID passport (a passport with a medicinal treatment and food chain status record but with no breed or pedigree record) by a passport issuing body operating under the EU equine identification legislation it can also be issued a registration book (a book with a breed and pedigree record) by an independent breed society or registry that does not operate EU equine identification legislation.
Outside the EU
Because a TCR registration book contains a Traditional Cob's microchip number as well as images of the cob, where a TCR registration book is issued for a cob that is outside the EU the TCR registration book could also be used as a general (or even official) identification document unless the cob is in a country where equine identification is regulated in a manner that would exclude a TCR registration book from being used as a general (or official) identification document.