Joomla loads a site's template files from a folder named after the template: /templates/<name>/ on Joomla 3, and /media/templates/site/<name>/ on Joomla 4 and 5. We find that folder in the homepage and, when the site serves it, read the template's templateDetails.xml, which gives its name, author and version.
Many Joomla sites are built on a template framework or page builder rather than a one-off design. YOOtheme Pro sites all use a folder called yootheme, Gantry 5 templates start with g5_ and JoomShaper's Helix with shaper_helix, so for these we name the framework: the design on top was made with it.
Limits: many sites block templateDetails.xml, so often all we can show is the folder name. A template can be renamed when it's installed, and Joomla's admin templates never show on the public site.
Enter the address above. We read the template folder the homepage loads its files from and, when it's public, the template's own details file.
YOOtheme Pro is a page builder that every YOOtheme site runs as its template, whatever it looks like. The design is built inside it, so YOOtheme Pro is the template you'd need.
Joomla's own default templates: Cassiopeia since Joomla 4, Protostar in Joomla 3. A site on one of them uses the template that comes with Joomla.
Yes. There's no sign-up and no limit on lookups.