In some cases, like with Mozilla's Firefox mobile browser (also tested in Chrome), you can request that the "Desktop site" be shown; that workaround allowed me to choose different songs.

There are also a couple of pages that seem to have a copy of the com.earslap.touchpianistapp APK, but I can't vouch for any, as I don't have a disposable mobile with which to test at the moment.

Open up developer tools, choose "toggle device toolbar" in the top left (or equivalent in other browsers), pick a tablet device, and set orientation to landscape. This lets you simulate touch events with your mouse.

I'm using a Microsoft Surface and had the same problem when trying to click using the trackpad, but tapping it on the touchscreen worked.

That works in Safari, but not Chrome for some reason.