Batteries can be confusing – there are so many variations and unless you understand the details, it is easy to get the wrong one.
Many LED hoops use Li-on 14500 3.7v 900mAh batteries. Let me break that down so it makes sense.
1) Almost all Li-on (lithium ion) batteries look generic because they are not mass market items. As you have noticed, you can’t buy them in stores.
2) 14500 indicates the size and layout of the battery. 14500 is the same size as AA batteries. But there are some high-capacity ones that are a bit longer (but aren’t designated 14650, which is actually quite longer), so be careful to check.
3) The voltage (v) is a “nominal voltage” which means it will vary – a little higher when the battery is full charged and a bit less when it is low. 3.6 and 3.7 are effectively the same. I have a suspicion the actual voltage is somewhere in between and the manufacturers are rounding up or down. *
4) The miliamp hour (mAh) tells you how much power is stored in the battery. A larger number (1200 vs 900) gives you more power and a longer spin time before the battery dies. But mAh over 2100 is usually impossible for the 14500 so don’t even bother.
BONUS: Chargers. Older Li-on batteries cannot be quick-charged. It will kill the battery and may even make it explode. There are quick-charging (45 minutes) batteries and they should be marketed as such. Since it is a challenge to know what is what, slow charging (2+ hours) units are never wrong. You can charge your 14500 batteries singly or in multiples. Chargers may accept a range of battery sizes, or only a single size. Make sure you get the charger that fits your size (14500) but mAh is not a concern – it will charge until full no matter the storage capacity.
(*) There are different chemical formulas for Lithium Ion batteries. Some go up to 4.2v (fully charged) and as low as 2.5 (discharged). They are still sold as 3.6v or 3.7v. For hooping, I think this doesn’t matter at all, but I am letting you know because it is confusing.