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Just seconding Chris. There's no reason to reproduce the site. You're question gets right into why, too—you wouldn't have a lot of control over which would rank. 301ing the exact match domain to the branded one is what I'd do.

Yes, they have keywords in Meta Title, but to me it looks more like keyword stuffing - it just repeating keywords in different language. When you look on the other competitors they usually do much better content wise and social media wise. It seems that age of the website is an important ranking factor here. Pioneer privilege?

Would you say that this just have "psychological" impact on the customer?

No problem, glad to help!

Thank you very much for this answer!

The structure is really going to be up to you. The differences in terms of what helps ranking are so very small today that it is better to choose based on user preference and your own. For consistency, it would be best to get a .co.uk that is on brand, but that route takes a ton of work over time. My favorite route is a subfolder (/uk) off of a gTLD in most instances. If your .com.au is already ranking well, expect that to keep happening until the UK subsite has had a chance to strengthen. And make sure your content is different!