No, Squarespace and Square are two separate companies. Squarespace is a platform for building and hosting websites, portfolios, blogs, and online stores. Square is best known for payments and point-of-sale systems, and it also offers invoicing, payroll, appointment booking, and Square Online, its own website and online store builder.
So there's real overlap, since both can run a website and an online store. The biggest difference is the cost to start. Square Online has a free plan that lets you build and launch a site and pay only when you make a sale, though free sites carry Square's branding. Squarespace has no free plan, so publishing a site means choosing a paid plan after the 14-day trial.
Some businesses use both, with Squarespace for the website and Square for taking payments in person.
The two can also work together. Acuity Scheduling, Squarespace's booking tool, accepts payments through Square, along with Stripe, PayPal, and Venmo.
When August Cura designed branded booking templates for Acuity Scheduling, the two names were sometimes mixed up. Some service providers booked their appointments through Square instead of Acuity, so a Square version of the templates was made to fit their booking pages. If you're sorting out which name means what, start with what you're building. For a website, Squarespace is the platform, and every August Cura website template is a Squarespace template.