Sure, but there are multiple ways to get there. They could have kept the LogoJoy site as it was and made a separate site/service called "BrandJoy".
You're hired!
Seems like it would work to just have LogoJoy as a product with its own page, and expand to Looka as the company name with additional products. Does anyone with experience know if that is done in the industry/would work?
It seems Automaticc did something like this. Wordpress was founded in 2003, and the company Automaticc in 2005, meaning they probably spun Wordpress out as the product. Now, the parent company owns a bunch of different products. Obviously speculation on my end.
Google re-organized with a new parent company (Alphabet) to do other stuff. They didn't change the Google name, so most users are blissfully unaware of the change.
I work in ecommerce/design/branding, and with very very few exceptions, mindshare and domain authority are always worth more than whatever ephemeral benefits you'll get from rebranding.