Yes, in most cases you can run more than one branded travel website from the same setup, so you do not need to build everything again from scratch, but how many sites you get and whether there is an extra charge depends on your plan, so please confirm that part before you commit to a second brand.
Each site can normally have its own domain name, its own logo and colours, its own home page content and its own contact details, so to a customer they look like two completely separate businesses. Behind the scenes you usually work from one back office, where bookings from both brands come into the same place and can be filtered by which site they came from. You can also keep your markup rules separate, so one brand can sell at different margins to the other, and staff logins can be limited to just one brand if you want your teams kept apart.
There are a few honest limits. The inventory you can sell depends on your supplier agreements, and some suppliers tie those agreements to one company or want to approve each website and domain separately, so your second brand may not automatically have everything the first one has. Payment is similar, because some payment providers want a separate merchant account for each brand name that appears on the customer's statement. Marketing, search rankings and reviews also have to be built up separately for each site, since they are seen as two different websites.
If you are planning a second brand, write down the domain names you want and ask us which parts of your plan are shared and which need to be set up again, so there are no surprises later.