Hello changadada,
While you can rename the default profile, you can't set another profile as the default:
You can change the name, symbol, and color of any profile, including the default profile. But there can be only one default profile, and you can't make another profile the default.
You can use the steps below to open links for specific websites using a particular profile, for links opened outside of Safari:
By default, links from outside of Safari open in the profile used most recently. But you can make links to specific websites open in a specific profile. For example, you might want to open links to your company's website in your Work profile, or open links to your school's website in your School profile.
1. Go to the website that you want to set up for this feature.
2. From the menu bar in Safari, choose Safari > Settings, then click Websites.
3. From the sidebar, select Open Links With Profiles. A list of open websites appears on the right.
4. From the pop-up menu next to a website, choose the profile that you want to use for that website.
This feature works only with links opened from outside of Safari, such as links in Mail, Messages, the Finder, or other apps. The link opens in the profile you chose for that website, unless the website is already open in another profile.
You can also find that information on this resource: Use profiles in Safari on Mac - Apple Support
Hopefully that helps to clarify your options.
Take care.