Is there an anonymized credit card for hotel stays?

I have heard there are "prepaid" cards and stuff like that, but have not tried one. Do they work at hotels?

Pre-paid cards do work at hotels, but I am not sure how using one would help with anonymity. As @nate-eldredge noted in their comment, you need to provide some form of identification along with the method of payment when you check-in.

I guess the underlying question here is how do I minimize my "footprint" when staying at a hotel?

The very first thing you should do is subscribe to a VPN service (or use one that is offered by your workplace).

A VPN is a private network that provides a level of security and anonymity over a public network (such as the Internet). There are numerous services that offer this online; some offer lifetime subscriptions for a nominal upfront payment.

The point of this service is to make sure your online activity is not tracked, and if the public access network at the hotel is compromised (for example, someone is monitoring the activity on this network), your devices are somewhat protected as they are using a private encrypted channel when browsing the Internet.

Its not fail safe though; you should exercise responsible computing by making sure your operating system and applications are updated and you are well aware of any software installed on your device.

The next thing is hotel receipts or other paperwork. You can use a device like this:

(Amazon Link)

To mask any sensitive data. There are many such devices available on the market. A search for identify theft stamp should give you some options.

Payment and the check-in process is the least of your worries when it comes to identity theft or privacy. As a cost of doing business you have to provide at least matching names on identification and payment. Even if you elect to pay by cash, the front desk will still ask for your identification (in some hotels, they still insist on blocking an amount on card for incidentals, even if you will be paying by cash).

Even if you were to stay at informal properties (like an airbnb, timeshare, etc.) you may still be subject to identify theft. Once can argue since these places are loosely regulated (if at all), you have even less guarantees of any privacy or recourse should something happen.

Unfortunately in today's digital age, the primary responsibility is yours when it comes to your identity and the protection of it. Data breaches happen all the time and in the end no matter what the PR machines of these compromised companies tell you, you are the first person to suffer should someone compromise your identity.