Why don’t I have to refrigerate store-bought baby food but have to refrigerate home-made purees

There are two components to flavor: aroma and taste. The wonderful smells you get are the volatiles of the ingredients you have been cooking, and while they add a lot to dishes, aromas are just part of what makes food enjoyable. One also needs to worry about taste, and that may be the missing part. When I decided to improve my cooking three years ago I was mystified by flavors. I wanted to know how spices worked and how to use them better, but all the reading and experimenting did not help my cooking.

Then I learned the Remick Maxim (lower the temperature when possible) and the quality of what I made went up more than in the entire previous year. I then heard Thomas Keller speak about salting and things clicked. One has to get the salt, acid, and sweetness of the dish right or everything else amounts to nothing. Once I learned to pay attention to taste, spices became easier.

The next steps have been:

• Understand caramelization (the Maillard process or reaction), it is a great source of flavor.
• The main ingredient needs to be rich in flavor. Even something as simple as rice will make a difference. Try making rice with Basmati from India or Pakistan, where they age the rice before selling.
• There are a lot of underdeveloped recipes out there. Beware.
• Use spices in groups that traditionally grow in the same region. Cumin and coriander, basil and thyme.

How long can I safely store Bánh chưng in the fridge?

It depends on how good the bánh chưng was wrapped (before cooking) and if the bánh chưng is put into the fridge right after cooking and if the bánh chưng is already cut open. i assume that you have a halfway eaten bánh chưng. I would treat this case like regular cooked rice. A post on cooking.SE says that cooked dishes last 3-4 days in the fridge. StillTasty says 4-6 days. I just asked my mother. She said it will be fine for one week in the fridge if cut. You may have a look at the comments on the question "How long can I store cooked rice in the refrigerator?"

Can I preserve it somehow? I've heard that some people boil or freeze it.

Boiling would kill some pathogens. But since the toxins produced by the previously existing pathogens won't be destroyed, the bánh chưng would be still unsafe.

Freezing: I'm a student and hoard such stuff in my freezer :D Regarding food safety freezing is not problem. If you store the bánh for too long and/or don't wrap the pieces tightly enough the bánh will have freezer burn. I ate frozen bánh chưng which has been for more than a half year in the freezer. It tasted fine after steaming.
After defrosting you should either recook the bánh or steam it. I think you already knew this ;)

After unwrapping, bánh chưng can stay good for several days while a wrapped one can be kept for two weeks.

I think this statement sound quite plausible if the bánh chưng was wrapped very tightly and was wrapped in many layers statement and is stored in a fridge right after cooking. Bonus: The bánh was cooked in a pressure cooker. I guess, it's even sterile for some time o.o

some posts on online forums, where people said it can last even several months!

I doubt it, too. Did they really said "fridge", not "freezer"? o.o
_{(Oh, or their bánh chưng drowned in lard. Lard lasts for a very long time x))}