72 hours visa policy

No, definitely not. A round trip is not and cannot be treated as any kin of visa free transit because you miss a third country. Every round trip requires a visa obtained in advance. However, you can adapt your itinerary to fit with the rules by introducing a third country, South Korea, for example. Like this you can create an eligible itinerary to look like this:

Tokyo - Seoul - Shanghai - Tokyo

Like this, the immediate points before and after Shanghai will be in two different countries and you will be eligible for 144 hour visa free transit in wider Shanghai area. As you can see, it is 144 hour visa free transit and not 72. 72 was replaced in Shanghai years ago in favor of more generous 144 hour visa free transit.

Thank You for your response and wanted clarification

1. We would leave Tokyo for Shanghai and fly direct with either AirChina or Japan Airline and arrive April 5th. We would fly back April 8th but take a different airline such as Asiana that will fly to Seoul and then Tokyo via a connecting flight (both Asiana). Would that count as a transfer and a 3rd country?

2 I read that an arrival/departure card is needed. Is that the one issued by the airlines or when we arrive in Shanghai?

Thank You for your response and wanted clarification

1. We would leave Tokyo for Shanghai and fly direct with either AirChina or Japan Airline and arrive April 5th. We would fly back April 8th but take a different airline such as Asiana that will fly to Seoul and then Tokyo via a connecting flight (both Asiana). Would that count as a transfer and a 3rd country?

2 I read that an arrival/departure card is needed. Is that the one issued by the airlines or when we arrive in Shanghai?

3 I read this from a website. there is an application process when arriving in shanghai. Any insight on this?
The application process for a 144-hour visa-free transit in China is as follows:
Gather the required documents. The list of documents is in the section above, “What Documents Are Required for a 144-hour Visa-Free Transit in China?”.
Notify the carrier when boarding that you will be using the 144-hour transit without a visa.
Complete your arrival and departure cards. The cards are available on flights.
Apply for the 144-hour stay permit. Submit all the required documents at the counter for 144-hour visa-free transit at the arrival port.
Claim your luggage, go through customs, and leave the airport.

Applying Online
The online application is available only in Shanghai. The process consists of the following steps:
Register on the official website. Choose a language, then register or log in for the first time. To do so, you will need to have a personal account.
Complete the application form. Click on “New Application” and start completing the application form. You will need to provide your personal information, planned places to stay, contact information, and arrival and departure dates. Then submit the application, and you will get a QR code.
Go to the 144-hour visa-free area. After landing in Shanghai, you are required to go to the 144-hour visa-free area and scan your passport or the QR code. Then print the Arrival Card you completed online.
Apply for the stay permit

Yes, that is something different and your new itinerary will have immediate points in two different countries. In other words, you will be eligible for 144 hour visa free transit in wider Shanghai area. As you can see, it is enough that a plane has just a landing in a different country and that is the detail that counts regardless of the fact that you will just remain airside in South Korea. So, you are fine and ready to proceed with your plans.

Arrival/departure card is just a formality. This is a short form (piece of paper) that you will fill out on board of the plane when arriving in Shanghai or if they do not dispense it on the plane, you will get one at the airport. You should really not bother about this very simple formality that requires several very simple answers/data.

Application process upon landing is also a formality. I will explain...

The first thing you will do at your check-in in Tokyo is to announce that you will fly from Tokyo to Seoul via Shanghai using 144 hour visa free transit. You will present your flight tickets and here I recommend you to have your flight tickets in hard copy. I know, this is boring, but it makes life easier. After a short check, you will be allowed to board the plane.

During the flight or upon landing you will get your arrival/boarding card that you will fill out. After that, the first thing you will do is to approach the 144 hour visa free transit counter (desk) where you will formally apply for a temporary entry permit. This is for free and here you will present your flight tickets and additionally it is also advisable to present your hotel booking confirmation (first overnight only is perfectly fine) also in hard copy. After a short check, you will be granted a permit and allowed to exit the airport. After you have exited to airport you have right to explore Shanghai city and two nearby provinces of Jiangsu and Zhejiang all on your own.

I do not recommend relying on online registration. Let it become more used by other travelers because it is still not exactly user friendly, so it is better to use conventional way explained above. Hopefully even the online option will become easier to use and it will, in the end, make all the formalities done in advance. However, I have already told you, stay away from this option for the time being and proceed with the entire procedure in a "manual" way.