I am not sure about MediaMarkt, but I had smooth experiences with €10 O2-brand prepaid SIMs from post offices and train station shops. Family members and I activated several in September, 2019.
EU cell phone service providers are now legally required to identify prepaid customers. O2 uses PostIdent, and the process can be completed entirely on a mobile phone, over WiFi.
Download the PostIdent app before you start, note the hours for live service (until 22h00), and visit the "freischalten" (unlocking, but in this context, really activation) link on the back of the O2 envelope. On the O2 site, give a German address, such as your hotel address. It is not verified. During the PostIdent video chat, give your home address. It is not verified either, if you use a US passport, which bears no official address. Both stages -- O2 registration and PostIdent verification -- have English options.
After verification is complete, return to the O2 site and try to log in with your new cell phone number, which is your identifier for most but not all activities. O2 will then send an SMS text message with an assigned password. Now, you can choose from different plans for voice calls, text messages and data. The €9,95 smart phone plan, covered by the value of the SIM card, provides unlimited EU calling, EU texting, and 1,5 GB of EU data for 28 days. Before you choose, review the plans in both the phone and Internet categories.
If the plan you want costs more than €10, add value to your SIM card first. The process takes you to a different O2 site where you create a separate account, identified by your e-mail address and a password of your own choosing.
You can add fixed amounts such as €30 or €50. US credit cards worked for me, though be prepared to contact your bank in case the first attempt is declined. If all else fails, many stores sell O2 vouchers. Some stores display cardboard cards with the O2 logo and different cash values, for you to show the cashier. In other cases, you have to ask the cashier.
Be sure to check the active plans, back in the O2 account identified by your phone number and the assigned password. It is wise to "abbestellen" (cancel) a plan that features automatic renewal, to avoid draining funds from your account. Rather than cancelling a plan "sofort" (immediately), look for the "Laufzeit" option, which keeps the plan in force until the expiration date.
You can keep your SIM card and phone number active by adding value periodically. I think the deadline is 6 months, but I am not certain.
Before returning to the US, remove the O2 SIM. O2 offers worldwide roaming, but it's so expensive that a megabyte or two of data used during an inadvertent connection in the US could drain all the leftover funds from your O2 account.
I do recommend keeping the card active if you might return to Europe. Having a known EU phone number provides peace of mind. For example, you can quote it when making hotel reservations, use it to register for local transit ticket apps (e.g., HandyTicket in Germany -- register before you leave Europe, as you need to receive your passcode by text message), and add it as an extra recovery number for your Apple ID, Google account, etc., in case you ever lose access while in Europe.
Bonus: by law, EU cell providers cannot charge extra for roaming within the EU.
Caveat: Mind roaming chargers when passing through non-EU countries like Switzerland by train!
Other bonus: O2 had a special this fall offering 150 (!) GB of free data, within Germany only (free, so not subject to the EU roaming rule, I imagine), upon SIM card activation.
Overall, I found the O2 setup process least onerous, and the prices excellent. I ended up using my O2 SIM card throughout Europe instead of buying cards for each country.