Is there any option or method of renaming my subdomain?
Right now it appears as merchandisemuseum.kesug.com on Google but I want to change it to MerchandiseMuseum.kesug.com, is there a way for that without creating a new subdomain and for free or not?
Subdomains are case insensitive; https://merchandisemuseum.kesug.com and MerchandiseMuseum.kesug.com are the same, which you can see when you visit both addresses as they link to the same page, yours. When trying to visit a (sub)domain with capitalization, most browsers will automatically convert the address to lowercase.
If you want a completely different subdomain, you can create it and park it on top of your current one:
What I truly meant was that it shows up as merchandisemuseum.kesug.com as the title on Google search and I just wanted to know that if there is a way to make it MerchandiseMuseum.kesug.com for the title.
But if your answer is the same then thank for the info.
Maybe in Google Search Console you’ll find something. If you haven’t already registered your subdomain there, you should. Last time I checked, it was possible to register free subdomains in GSC (but not AdSense).
Domain names are not case sensitive, and in most cases they are always shown as lowercase only.
The title of your website in Google can be changed by setting the og:site_name meta tag on your pages. Google will use these to tweak how your site is displayed in search engines.
Ok how and where can I change or edit that?
beacuse now it appears as " kesug.com " above my full URL https://merchandisemuseum.kesug.com on google search results, it there a way to edit that?
The short answer is: you can change it in your website. As for how you change this, or anything else, in your site, you know that better than we do.
A quick look at your site tells me you’re using WordPress. If so, the best way to fix it is probably with a WordPress plugin that configured the OpenGraph tags for you.
I suspect SEO plugins like Yoast or Rank Math have this functionality.
I think there is a misunderstanding,
Let me give an example :
When you search Amazon you are presented with a blue highlighted title and the meta description below it. I didn’t mean the snippet because I already have that edited using my rank math plugin.
What I meant is the title above the url in case of Amazon the url is, https://www.amazon.com and the title is just Amazon, but for me the url is https://merchandisemuseum.kesug.com but the title is kesug.com. I hope you understand that I meant the title above my website snippet NOT the title within the snippet itself.
For this issue You need to get Custom domain… not subdomains… thats the only way…
Yes, there is. I am telling you the solution to your problem, which is a problem I clearly recognize, and know how to solve. Yet you insist that I don’t understand, and are refusing to attempt the solution I present you.
I completely understand what you mean, and have done so since your first message.
So now I hope you understand that the information I’ve been telling you is the solution for your problem.
That kesug.com thing in the search results is the site name. You can set the site name with OpenGraph tags on your website. If no such site name tag is set, it will show the registered domain instead.
There is no hosting setting that determines this, we cannot set this for your site, and you also cannot set it through Search Console.
If you don’t believe me, then please believe Google themselves:
Ok understood and sorry for my insistence.
Thank you so much for the information 
.
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