Is this a scam or is it legit?

Have you looked at the Ryanair website to see what the "contact us" options are?

To be safe I would use what you find on the website, vs. clicking on a link in whatever was sent to you until you confirm its authenticity

No personal experience

I would also contact Ryanair using the info on their website. Sad to think that there are so many scammers these days, but better to be safe than sorry.

Do you at least have a record locator number? What documentation for this flight purchase did you receive from Kayak / Booking.com? I would not respond to an email, or a link in an email, I would call Ryanair.

The general consensus here is to avoid third- party booking sites for airfare. If a flight gets canceled, or other problems occur, it's very hard to get resolved. The third-party booking site sends you to the airlines, and the airlines send you to the third-party booking site. Perhaps this is a blessing in disguise, if you can just get out of the transaction.
Good luck!

Have you checked the fares on the actual Ryanair website? Ryanair doesn't like third party bookings. If you can get out of this and book directly with Ryanair you will be better off.

It's also a good idea to peruse the Ryanair website to learn all the rules of flying the airline or it's going to cost you a lot more money.

The Ryanair website has info about verifying third party bookings. Here is the FAQ.
https://help.ryanair.com/hc/en-gb/categories/12489248662673-Travel-Agent-Bookings

The email is probably legit, but I would make sure that you verify through the Ryanair website.

Ryanair operates on a direct to customer distribution model and has long battled third party vendors. This article provides some background -- https://simpleflying.com/ryanair-swiss-court-ruling-no-impact/

I will just say it. You should have booked directly with Ryanair. Second, you should have chosen (if even available on Kayak) a higher level of service, specifically the "Plus" option which allows airport check in, a 20 kg checked bag, and a small personal bag on the plane. Anything else, you will be back here complaining about how terrible Ryanair is.

Pray that your ticket is not for "Basic" fare, it will not work for you. Even "Regular" fare likely will not.

Edit: I will add, I like Ryanair, they are my go-to airline for budget hops in Europe, I have had excellent service and satisfaction with them, you just have to know what you are buying.

Ryanair wants to know exactly who you are and how to contact you. You need to go through their process before you can check in or manage your booking. It is legitimate.

This is why I always use Google Flights to find flights especially in Europe. You can identify the best intinerary for you, save it to get notified if the price changes, see the price trends, etc.

If you do a round trip, you can put in your dates, get the results and then click on Dates to see a spreadsheet with your departure date across the top and your return date down the right side. I find this very useful to see how prices vary during the week.

You can click through to your chosen itinerary to get the details and click through to purchase directly from the airline. That's not going through a 3rd party. Or you can go directly to the airline to book, armed with your detailed flight information.

I did a test choosing Ryanair in September and saw that the RT Rome to Athens flights are very cheap. I also saw this very useful info about baggage, shown before the option to click through to purchase the ticket:

Ryanair allows passengers to board with 1 small item, such as a purse or laptop bag. Passengers who pay a fee may also bring 1 carry-on to stow in an overhead bin. The fee for checking your bag is €/£10-18 for bags up to 10 kg, and €/£25 + €/£10 in high season for bags up to 20 kg, for each flight.

There's also a link called Ryanair bag policy that takes you here: https://help.ryanair.com/hc/en-gb

When I clicked on the bag policy link, it goes to lots of info, the most important of which may be under Bag Rules.

The last time I flew Ryanair was in the summer of 2019. It was one-way from Porto, Portugal to Barcelona. When I bought my ticket, it was so cheap that it was well worth paying for the aforementioned:

Priority & 2 Cabin Bags: This allows you to carry a small personal bag (40x20x25cm), which must fit under the seat in front of you and a 10kg bag (55x40x20cm) onboard, to be stored in the overhead locker. It also allows you to board the plane first using the Priority Boarding queue at the gate.

It was astonishing how many people not only didn't pay in advance but also came to the gate with enormous bags. I can't imagine how much they had to pay to check them. Their line was very long and I saw none that would be legal cabin bags.

I hope you'll be able to sort things out for your flights, including the potential baggage issues, using the link someone else provided earlier.

Skyscanner, Google Flights, Kayak etc will search for flights and direct you to booking. Best if they direct you to book direct with airline. However, the best price will often be through a third party site like Expedia or booking.com. I booked our British Airways flights for Portugal this way via skyscanner and expedia. I could not get the same combination of flights at the same low price on the BA site. I had to provide our passport info via Expedia to buy the tickets and then had to do the same on the BA site. I don't know how it works for Ryanair. Good luck.

Thank you to everyone that responded. We are not frequent flyers and it's our first time to Italy and Greece, so this is a learning experience. The email was legit and we did go to their website to verify and not the email. We'll probably experience hangups with our luggage even though I paid in advance to check them (I think?) they may be oversized. What a pain but I know it will be worth the experience. Thanks again to all!

By oversized do you mean larger than the 31x47x47 inches? ( for a checked bag)

Or over the weight limit you selected. They offer 20 or 10 kilogram options ( 44 or 22 lbs.)

Which did you choose?

Kayak.com is a useful research tool, but I only use it for that. I never book through Kayak.
I always go directly to the company.

I mistakenly booked a hotel reservation in Rome for a Hilton Hotel that I thought was the company website. Instead it was a third party website that for some reason google had it first on my search list. Now, I pay attention more closely when I book.

I later cancelled my trip to Italy and when I cancelled this hotel reservation, there was a $40 penalty.