Only if you try to charge it from 0
I would suspect in order to get the "same battery life as original" they are probably overvolting a little bit and using an inferior cell, which means the battery discharges much closer to "the battery is actually, truly dead 0%" than the stock battery does.
If the voltage gets too low, there is some small risk in charging it. That initial "recovery" charge can generate a lot of heat in a hurry.
So I strongly suspect when your battery is at 0%, your phone is saying "No, I'm not throwing power at that, lest it go all Thermite on me - that there is a dead-as-a-doorknob battery".
The external charger is obviously not as fussy.
If that is correct - and this is not unusual for an eBay cheapo battery, I would strongly recommend you never, ever run that battery down to "indicated 0%" again. Or if you do, always charge it with the charger they included, and avoid leaving it unattended or near flammable objects.
The battery will *probably* work just fine as long as you don't fully discharge it ever again, and watch it carefully as it nears 100% (the BMS might be under-reporting that so it overcharges for a little extra battery life, and again - heat - thermite - bad things are unlikely but possible).
Or, for 11 bucks, it might be advisable to look at the cost of a replacement phone if it fails in a bad way as compared to the price difference of a battery made by a more reputable manufacturer that stands a much lower chance of failing. Your phone, your battery, your decision to make. I'm advising, not judging.