Which technologies should I consider when building a platform similar to AirBnB, eBay, and Amazon?

PHP is not outdated, there is no real concensus on what is the most popular programming language out there, some people are obviously biased, some based it off number of job postings in which case PHP and possibly .NET are the top ones. It depends on what you are doing. See them as upgrades on each other.. not that they are bad, just depending on the workload you'll put it through.
With that said, there is no real answer to this either, it depends on who you can find, the number of resources you yourself can find on the language for help, feedback, etc.
Go to builtwith.com and it will tell you the stack any website is built upon including the framework and languages :)

I didn't follow some portions of your question, and hopefully you take this well :).

"I am also considering creating a native app similar to CakePhp or Node.js" --> There are three separate concepts in there that are worth unwrapping, so probably good to talk about that first:

CakePHP is a PHP framework, comparable to Symfony/Laravel/CodeIgniter/etc.

Node.js is a JavaScript language & runtime, comparable to PHP, Python, Ruby, .NET, etc. Similar to Cake/Laravel/etc, there are many great frameworks in Node.js; Express, Meteor as you mention (well, Meteor is more of a tapestry of many technologies, and it's awesome!).

Native apps are usually iOS, Android, Windows, etc, but can be hybrid, too (Ionic, Phonegap, React Native, etc). Connected native applications usually have a backend, which is usually written in a language like the ones mentioned above, and usually with a framework written on top of it to accelerate development.

With that out of the way, let's tackle the main meat of your question:

Can PHP be used to create a platform similar to AirBnB, Ebay, Amazon?

Answer: Absolutely!!! Just one recent example is Slack, who's backend is build on PHP (LAMP stack). Check out this tweet where they mention their stack: https://twitter.com/SlackHQ/status/426469205005705217. Another good discussion on it: https://www.quora.com/What-web-framework-s-does-Slack-run-on.

One bottom line is that choosing a technology stack is important, but more importantly is the correct and effective development on the chosen technologies. I've seen both beautiful and horrible applications written in most languages. Much care needs to be spent architecting and developing the solution, always using best practices, and considering what's the right decision for the business at hand (hint: there's no silver bullet).

My suggestion: choose whatever you're more comfortable with. Also, surround yourself with someone that has created something of the magnitude of what you're aiming for, and ask for advice.

Now, for a opinionated answer: I'm currently building most highly-scalable backends on Node.js + some kind of framework. The frontend is, in most cases, being built using a client-side Javascript framwork such as React or Angular - this is assuming it's going to be heavy client side, which means that one can spin up a native app using hybrid methods. There are many cases, though, where we're building native apps, too. Again, this depends on the business case for a particular choice.

Happy to jump on a call, show you some examples of this, and answer any questions you have!

I love the question "what technologies/frameworks should I use to build my new product?" because it is one of the only questions for which there is almost always one correct answer for everyone. That answer is 'whichever lets you get a minimum viable product released the fastest.'

To sorta-quote Eric Ries, most products do not survive their first encounter with the users. You should use whichever tech you are the most agile with and get a concept app out and into people's hands as fast as possible. After you pivot 15 times and actually know what your immediate and near-term requirements are...THEN you can start looking at going off into new territory if there's a better solution than PHP for your particular case. I'd be spending much more time trying to be sure that you are building something that people actually want and that you can create a business around rather than learning a new language. I'm happy to talk over strategy anytime :0)

According to the “Developer Survey Results 2016”, which was conducted by STACKOVERFLOW, among over fifty thousand developers from all over the world, PHP is more widely used back-end technology compared to Node.js.
In 2015, PHP 7 with new long-expected features was introduced. Possible differences in performance is already not an issue when one compares “Node.js” technology with “PHP” technology.
From the scalability perspective, there are no big differences between PHP and Node.js, it is more important to consider the project’s architecture. https://belitsoft.com/php-development-services/php7-vs-nodejs
In general,it is a good idea not to oppose them, but use them cooperatively.