Why are they popular? I've actually thought of that myself.
I know some people like to collect the ones they just to show their support. I only buy the smash bros series, and I only buy my mains. I currently own Sonic, Greninja, Sheik, Toon Link, and Jiggylypuff. These are actual characters I main in Smash 4 and previous smash bros games. They feel like trophies to me. I do not use them for their actual DLC purpose, tbh. Also, I also don't own a wii u or a new 3ds.
But, this eventually grows into me as I just want to collect more by temptation, and I just want to expand my collection since I already have 5 out of the 53 smash bros. For example, one of my friends only collected the Legend of Zelda Amiibos, but eventually it just slowly grew into an collection as he wanted the Mario Bros Series. Eventually, he just wanted every single amiibo, and he dedicates his time to trying to collect them all.
Some people only buy the because they're a "one time only" deal and the fact they are rare. For instance, one of my friends went to gamestop, and he saw a rare Shulk Amiibo. He has absolutely no idea who Shulk is or even played XenoBlade games, but he only bought the amiibo because he knew if he didn't, someone else would buy it and it wouldn't be there anymore. It's a one time deal even. The shulk amiibo just sits in his table unopen and forget he even owns it. Just the fact that amiibos are rare and limited makes it the more reason to want to buy them. Also, some people just like owning rare things, and it's bragging rights. I was bragging about my jiggylypufff amiibo to my friend that I owned it. I'm heartless, I know. But he owns 40/53 smash bros Amiibos, and I just wanted to brag that I own one he doesn't! xD
Also, I bet most people that own the Lucina or Shulk Amiibo, when it came out, never even played Fire Emblem Awakening or XenoBlade. Some people just like to collect them just to re-sell them, or like to own it just because it's a rare. Scalper or re-sellers like to buy as many as the new wave amiibos as possible just to sell them online for a profit. Unbelievably, it's an easy cash grab because they only have to wait in line for a couple hours and pay 14.99 for each. They go to each TRU, Target etc and take as many as possible. Sometimes, Scalpers would bring their Aunts, Parents, children, anyone they can to help them get more because its "one amiibo per customer". Then, they sell them for around 30 to possibly 80 dollars. I waited in line just to get a Jigglypuff and Greninja, and I paid 14.99 for each. However, some people have no choice but going on amazon and pay around 30 to 40 dollars just to get a Jigglypuff or Greninja amiibo.
Lastly, some people just like to own them just not to feel left out. I can't tell you how many of my friends in our little smash group just bought a random amiibo just so they don't feel left out. Eventually, it grows into them as they want more and more based off the other reasons I mention.
Tbh, I wish they weren't popular because just getting the amiibo you want is a pain. You have to wait in stores as early as possible during a school day. Some people wait from midnight just to get the ones they want. It's also annoying because people cut in line, and others are aggressive especially if they do not get what they want. Also, the fact that waiting in line isn't enough to get an amiibo. Some of them only have 1 to 3 in stock. So, there a possibly of being disappointed and wasting your time.
Buying them online is even worse. For example, the lucina amiibo on Amazon was on sale and ONLY LASTED 5 SECONDS. It was so bad, some people did not even get the sale page. The Ness amiibo on Gamestop was also bad as so many people were trying to refresh the Gamestop site that IT KILLED THE SERVERS. They had to cancel it.
It's so ridiculus to pay lot just for these amiibos when some of them look really bad or look nothing like them. Also, the painting isn't soo good, or some of them lack detail.
Overall, they're popular for lot of reasons, and I believe the fact that they are rare is what makes them popular.