Are Apple's AC Cords for MagSafe (1+2) always the same?
Absolutely no problem to change the cable!  Any Apple adaptor cable with a similar connector since the last decade will do fine.  Even the plugs from your iPad charger will work.

In fact, you also can use a standard radio cable (sometimes called an 8-shaped power cable).  You can find them anywhere, in shops for $2 but probably also attached to various household appliances.  Works fine.

As you can see, your Apple adaptor cable doesn't have a real 8-shape, but rather a B-shape.  This makes the cable useless to attach to your radio, but the radio one can be attached to the Apple adaptor.  There is no obvious reason for Apple to do that.

We regularly buy/replace Macbooks while traveling.  Therefore we end up with exotic power plugs without too much use.  Rather than using a travel adaptor or replacing the whole cord, we just cut off the plug-side and solder or screw on a EU plug bought at the local radio shack.

Cutting the cable and placing 3rd party plug will obviously void the warranty on that very cable, and might be illegal according to your local law.  In Germany this is allowed, as long as your new plug is DVE-certificated.  In any way, it's 220v and placement should be done with care.