If I'm understanding your situation correctly, your old domain is still indexed in Google. When a user clicks on a result from your old domain, they are 301 redirected to the page on your new domain.
It will take some time for Google to convert all of your old domain pages to your new domain pages.
This is because Google has to crawl every single page on your old domain in order to realize that page has been 301 redirected. That's just how Google works.
As Google crawls the indexed pages on the old domain and sees that it has been 301 redirected, it will reindex those pages as results for your new domain.
How long it takes for Google to crawl all of the pages on your old domain depends on a lot of factors. How many pages are on your site? The more pages the longer it will take. Is your site crawled frequently by Google or not as often as you might like?
It will take a while for Google to crawl all of the pages. But as it crawls more and more pages and discovers all of the 301 redirects, it will pass your link juice and domain authority over to your new domain with each crawling update. It will pass your link juice and index status one by one as each old page is recrawled.
Your new domain might not be ranking for the search term of your brand. And as Google crawls your old pages and updates the 301 redirect your domain should start ranking higher and higher in the search. But getting your domain to the top of search also depends on a lot of factors. Now that you've switched domains and brands, there's no guarantee that you will rank #1 for your brand term anymore.
If your new domain brand name is a competitive keyword, it may be difficult to rank for it. If it's a random word that does not have much competition, it may take a little while for you to rise to the top.
Part of Google's algorithm for ranking a brand's search term is based on whether it believes you are the #1 source for that brand/word. As it discovers your new site more and more you should begin to rank higher again. And this will come in time once all of the 301 redirected pages have been updated.
If your old domain was a powerful brand with a lot of users, those users were probably searching for your brand and clicking on your result in Google. This pushed you to the top of Google rankings. Now that you have a new domain and brand not as many people are searching for you in Google and clicking on your result. Because of this, Google has not yet determined that you are #1 for your position.
All of this may just take some time. But you'll most likely get there if everything has been done right.