Do I need bring my own car phone mount for rental car in Greece?

What type of phone holders is the best for the purpose since I need mount it and later detach it?

I've rented dozens (maybe hundreds?) of cars in countries all around Europe and around the world. Only once - just one time - was there a phone-mount in the car when I picked it up (that was in the Faroe Islands).

No, don't count on there being a car phone mount in any rental car, anywhere. Bring your own.

Nowadays I always bring a car-phone mount with me if I think there's any chance I might be renting a car. I've also got a roll of duct-tape in my emergency fix-it kit (that always comes with me on any trip) which can work in a pinch (it's ugly, can leave sticky goo on your phone and the car dash, but hey, if need it you need it...).

Edited to add: Small bungee cords can also work in a pinch, if you happen to have a couple with you (I always do), but it may be tricky depending on how the car's dash is configured. I just reviewed photos of our Azores trip (where we rented 5 different cars so we got pretty good at figuring out how to mount the phone with no real "mount"!) and it often worked out that the combination of bungee cords plus a bit of duct tape was the best improvised system I was able to come up with (of course, it depends on what you have to work with). 😎

Here's a photo of one solution: https://adobe.ly/3FIrm89
This was the 4th car we rented on that trip, so by that point we had refined the setup (no more big wads of duct tape) for probably the "cleanest" jury-rigged "mount" we came up with on that trip. Worked great!

I brought my own phone holder. It fit in the air vent. I brought my own charger cable too. The rental car had CarPlay, So when I plugged in the USB cord, the Maps showed up on the rental car video screen.

What type of phone holders is the best for the purpose since I need mount it and later detach it?

I would get one with some kind of suction holder. You could easily find on on Amazon. But I agree with the others that you should bring one just in case. My last 2 cars did not have them. Some cars will have Carplay, but again, not all of them.

suction holder may leave some adhesive on board. Then car rental company can charge you for that.

A suction holder is not adhesive. It is just a suction cup. It might perhaps leave a faint ring on soft plastics, but I would not worry about it.

I think suction holder still have some adhesive at the bottom. At least my soup holder in bath room. So it may still be a way for rental car company to charge me.

I think rental companies make money mainly from selling insurance and extra charges rather than the rent of car. Just like Costco make money mainly from membership fee.

https://www.amazon.com/WeatherTech-CupFone-Universal-Adjustable-Portable/dp/B07KY1XKRQ/ref=sr_1_1_sspa?keywords=weather+tek+phone+holder&qid=1698957358&sr=8-1-spons&sp_csd=d2lkZ2V0TmFtZT1zcF9hdGY&psc=1

This type will not leave a mark

Not all cars have a cupholder right there. Not all cupholders are round.

As someone who has spent years looking out the front windshield (of both cars and airplanes), while referencing maps on screens, I can tell you with some authority, it's really not a good idea to be glancing down at your knees (or something just to the right of your right knee) for map details while zooming around unfamiliar places.

It's dangerous enough simply trying to divide your attention between the world outside and a small screen inside, as you're moving through the world. But if you're going to do that, it's less dangerous if you can at least position your little screen as close to your driving line-of-sight as possible, so the glancing back and forth is less of a context switch.

I prefer my maps mounted up near, but not blocking, my view out the front windshield. It may involve some compromises and it may not be perfect, but looking down at where a cupholder would be scary dangerous for me. To each their own (but make sure you have good medical insurance, maybe with repatriation benefits if you're going to be constantly glancing down at your cupholder as you drive around...). Good luck and be careful.

Agree with periscope!

This multiple trip planning is sounding way more fantasy than reality
but lots of folks along for the ride, anyway.

Why do you say that? I just got back from 5 weeks in Greece and Italy. (the same places the OP is traveling to) .I rented a car from Athens to Meteora and visited many of the places the OP is considering visiting in Rome and Italy and Greece. Seems like a reasonable trip to me.

I am a Type A person, so thinking about the details is what it is all about. There's nothing wrong with that. FWIW, in my planning, I actually viewed the street route on Google Street View from the Car rental place on Syngrou Avenue to the freeway onramp just so I could see what the road was like in the city center area. I also google mapped the walking route from my hotel to the car rental place so I could decide if I needed a taxi or not. (I didn't).
Clearly the OP doesn't have as much experience renting a car either here in the US or abroad, so I guess it's a fair question. The issue of extra car insurance is an issue that all car renters have struggled with at one time or another and some of us have found solutions that we would like to share with the OP.

After the OP's trip, I hope he/she will write a trip report and share what worked and what didn't, what he/she liked or didn't like, what he/she could have been done different or better, what was over planned and what was under planned, but I doubt that will happen. (some people don't have the self-awareness to do so). It's also tough laying out your travel disappointments and having the people on this forum blame you for your lack of planning (i.e. see the comments on bostonphil's trip report on her trip)

Note that some countries require that the phone be mounted below your eye level. Make sure the cigar lighter fuse is good when you first plug in the phone.

My experience is that most rentals have these days have Apple Car Play installed in them. So you just need to plug your phone into the USB, and Google Maps or Waze will show up on the car's built in display.