My Galaxy Tab 3 will only charge from the wall adapter from Samsung I'm going on holiday soon and we planned to take a USB hub to charge all of our devices rather than take several wall adapters, however when I checked that my 7" Samsung Galaxy Tab 3 would charge, I found that it didn't. After investigating, I found that my iPad charger wouldn't charge it either and in fact the only that would charge it was the USB wall adapter that came in the box with the tablet, even though the port on the USB hub and the iPad charger are rated at 2.1 A (5 V).

When I plug the tablet into to a non-Samsung adapter powered off however, the screen flashes lighter and darker but nothing else happens which seems odd behavior to me.

What is the reason it will not charge by any other means, and is there a solution?
There is more to an adapter than just the current, although kudos to you for knowing that much.  The problems arise due to the fact that the USB-2 spec is for a maximum of .5 A at 5 V without some extra handshaking to ascertain that the device wants and can handle more.  For Apple devices, this is done in a certain way according to their "standard", which is different than PCs.  So, for a PC, you normally need a driver, and it sets up the port for the correct voltages.

Most aftermarket USB chargers are designed to work with Apple devices, and seem to have some differences that make them either more or less compatible with Android devices, but usually require what's called a "charge-only" cable or adapter, that shorts together certain pins of the connector to make the device charge at the higher rate.  These can be bought on Amazon from companies like Mediabridge.  But it's not 100% certain if any particular adapter will work with a particular device.  Unfortunately, some Samsungs won't work at all with some chargers, and work fine with others (I have an 8" Galaxy Tab 3).

If the tablet recognizes the charger at all, it will probably work with the charge-only cable/adapter, I think.  I have one charger where it doesn't even register that it's connected, and it won't work no matter what I do.  But it does work with most adapters, including the one I use for my wife's iphone.

With the tablet powered down (not standby), it should charge at .5 A, which will take much longer.  I'd try the charge-only adapter if you have time to get one before you leave.  Otherwise, you may just need to bite the bullet for now and bring an extra one for the tablet.