I have an old site called for example domain123.com in health market Now the question is if we want to register domain like old one which is domain123.co.uk in cars market Is Google will make a relationship between domain123.com and domain123.co.uk in ranking and relevancy although the two domains have the same brand but in different markets ????
are you talking about redirecting one domain to the other? or are you talking about the same name but two different extensions and website content?
Google will not automatically associate the two domains. They know the difference in the on page content and niche
In basically, Google will not treat 2 websites with having any relation. Because they're difference in content, market, keywords, traffic, audiences,...and many other factors. The domain names are just a brand, and they can be same, however they have difference extensions so that Google will not see them the same. I think so.