It seems pet travel to Northern Ireland won’t be as straightforward from January 1st. 
Travelling to shows, importing and exporting puppies/dogs.

https://www.gov.uk/guidance/pet-travel-to-europe-from-1-january-2021#finding-an-official-vet

NI is part of the UK & therefore no longer in the EU. To travel to & from the UK from NI does not involve the Pet Travel scheme

What you have linked to involves the Pet Travel scheme for travelling to EU countries not internal UK travel

'The government has worked alongside the European Commission to ensure pet travel can continue between Great Britain and the EU and Northern Ireland at the end of the transition period.

From 1 January 2021 onwards, the UK will have Part 2 listed status under the EU Pet Travel Scheme, meaning that people travelling from GB with their pets and assistance dogs will need to follow new requirements in order to travel to the EU and Northern Ireland.

Before taking their dog, cat or ferret to the EU for the first time after 1 January 2021, pet owners must complete the following steps.

The only new requirement for travel to the EU is the use of a certificate, rather than a pet passport:
•Ensure their dog, cat or ferret is microchipped.
•Ensure that their dog, cat or ferret is vaccinated against rabies – pets must be at least 12 weeks old before they can be vaccinated.
•Wait 21 days after the primary vaccination before travel
•Dogs must be treated against tapeworm 24-120 hours before landing, if they are travelling to a tapeworm free country.
•Visit their vet to get an animal health certificate (AHC) for their pet, no more than 10 days before travel to the EU.'

New rules for pet travel from 1 January 2021

> meaning that people travelling from GB with their pets and assistance dogs will need to follow new requirements in order to travel to the EU and Northern Ireland.

Is that a typo because I can understand travel to the EU and Eire ..... not to Northern Ireland??

>> meaning that people travelling from GB with their pets and assistance dogs will need to follow new requirements in order to travel to the EU and Northern Ireland.
>Is that a typo because I can understand travel to the EU and Eire ..... not to Northern Ireland??

No, not a typo. The EU demands that the whole of the island of Ireland is treated as if it was all Eire. So rabies injections, tapeworming treatment and travel documents are needed for the very first time between mainland UK and NI.

The EU can demand what they like the UK is no longer a member of the EU so have no say in travel between the UK countries.

Tapeworming only has to be done if one of the countries has Echinococcosis multilocularis. Neither UK nor Ireland have it

>The EU can demand what they like the UK is no longer a member of the EU so have no say in travel between the UK countries.

The terms of  the Brexit agreement, to avoid a hard border between NI and Eire, is that the border is effectively in the Irish Sea, and in this respect NI will be treated the same as Eire - as an EU country.

>Tapeworming only has to be done if one of the countries has Echinococcosis multilocularis. Neither UK nor Ireland have it

The government website says "You’ll need tapeworm treatment if you’re travelling to Finland, Republic of Ireland, NI, Norway or Malta."

ah Brexit- the gift that just keeps on giving..

Sadly JG is right. Currently as it stands there is a bloody border down the Irish Sea. Talk about splitting the union up. They've had years to sort this sorry mess out and still they have come up with calamitous decisions which does no one bar the super wealthy any good.

Well hopefully it will stop some of the puppy farmers from coming and going across although I doubt it.

>Well hopefully it will stop some of the puppy farmers from coming and going across although I doubt it.

As the rules stand currently it won't make an iota of difference because there are no plans to change the rules on entering the UK - underage puppies will still come in on false paperwork. It's only leaving the UK to enter an EU country that is being made harder.