There really isn't anything out there like neopets.
I've recommended it before, but Goatlings.com is a site that reminds me of an early neopets. There's restocking, auctions, dress-up avatars, forums, etc.
And adorable pastel goats.
I also love the fact that the creator takes on a lot of user-based content, so if you design something, there's a chance it could make it onto the site. If you wanna take a look, here's is the link.
Sadly I'm not sure any other site would give me what I got out of neopets and loved so much, which is the sheer variety of it - if I wanted to play NeoQuest for a week I could do that, or I could try kadding or RSing or auction sniping, or KeyQuest, or flash games, or questing for the perfect pet customization items, or trying to collect a bunch of the same pet species or color, orrrr curating a gallery, or collecting obscure avatars, or Plushie Tycoon or or or! The sheer diversity of ways to play is what grabbed me back in the day and kept me coming back.
People always mention Subeta.
The thing about it is that while it is super easy to make SP on there, due to that, many items are super expensive. Even really common items are pretty pricey. And if one of those items just so happens to unlock and achievement it jumps in price ten fold. The games on the site are okay, you can buy cash shop money with in-game money (But its pricey, of course), you can also do commissions and get in-game/cash shop money/items that way.
The site is a huge popularity contest.
The primary focus of the site is on dressing up your avatar. With its focus being shifted far away from pets and only on human avatars, it just feels like a chat room for teenagers. It was far more like an early-era Neopets when it first came out.
Pets are secondary to that by a long shot as they constantly pump out wearable items more so than anything for pets. The pet art is really nice, however, and it is easy to get a coloured pet (Through zapping), but other than that you tend to forget to do anything on the site as there really isnt much todo unless you are a social butterfly then you can just hang out on the forums.
SO no, not a big fan of Subeta. It feels like it's not well maintain and chaotic. The freedom of naming is awesome but a line needs to be drawn. Apologies to those who like Subeta.