According to the Yellow Fever Frequently Asked Questions (YF FAQs) - International Certificate of Vaccination or Prophylaxis (ICVP) by the National Travel Health Network and Centre, UK, the passport information on the ICVP is required only to identify the ICVP holder in case of a name change:
Q. The front of the ICVP has a place for 'Passport Number or travel document number'. What does this mean?
A. Travellers can add their passport number on the ICVP if they wish. They do not need to bring their passport with them to the appointment unless they have changed their name and are requesting a duplicate certificate or a new certificate in their new name.
I would therefore assume that, unless you plan on changing your name, and if you do so you should probably request a new ICVP altogether, you are not obliged to update the passport information on the booklet. If you want to be on the extra-safe side, you could always request a new booklet, transfer your existing vaccines information, and update the passport details accordingly.
Requesting a New Booklet
Is is indeed possible to request a new ICVP. This can be done either at the centre where you were issued the first ICVP, or in any other authorised centre. These include most of the Yellow Fever Vaccination Centres, across the globe. Note that the former is the preferred method, as you can be sure that the centre will hold all the patient-related information concerning you.
Given proof of the previous vaccinations, these will be transferred to the new booklet. Quoting from the Yellow Fever Frequently Asked Questions (YF FAQs) - International Certificate of Vaccination or Prophylaxis (ICVP) by the National Travel Health Network and Centre, UK:
Q. Is it possible to reissue an ICVP?
A. It may be necessary to reissue an ICVP given within the previous ten years in some circumstances.
In either scenario, re-issue of the ICVP must be completed by a doctor, nurse or pharmacist working in a designated YFVC. Information about the original YF vaccination must be available (e.g. copy of patient record,
vaccination booklet, or communication from a health professional in the original centre).