We are researching a catamaran trip for 8 around the Greek islands in August for 5-7 days, being able to anchor at islands for a day or two. Does anyone have any experience or advice on this?

There are lots of yacht charter companies, mostly in the Athens ports area, that provide such a service. You may be able to find what you want on Paros or Syros, too. As they are very expensive, you need to do research beforehand and learn as much as possible. A good place to start is the independent website Sailing Issues, and its page called “The 8 steps towards a successful sailing holiday” is here:
Free yacht charters guide: 8 steps towards a perfect sailing holiday in Greece and Turkey.

A yacht charter for a group of 8 friends sounds like fun. For the record, I have always been interested in sailing, but my holidays in Greece have been land-based, taking ferries from island to island.

I am hoping that someone will suggest a 2 day 1 night sailing company that is not too expensive.

I know one that does a full week, and used them successfully 2 years ago: Archipel Club. The company is based out of Paros. Their prices range depending on which month you are sailing, and how big the sailboat/catamaran is. We sailed in September for a week with Captain Bertrand, who cooked wonderful breakfast and lunches. It came out to about $1000 per person per week.

https://www.archipelclub.com/index.html

My trip report:
The Little Cyclades, Santorini, Vienna

You also don't have to rent the whole boat if you can't find other people to join you. You can just buy one spot on the sailboat/catamaran and meet different people.

Not actually a "catamaran" but we had a wonderful sailing trip with Seascape Sailing. SeaScape Sailing | Welcome to SeaScape Sailing Great boat. Great crew. Great stops. Their website also includes a lot about how to choose. We were just a couple and joined others on a regularly scheduled trip.