Just to be safe, measure them both when fully charged, they should be the same-ish voltage (capacity aka. mAh doesn't matter). If they are different voltage they might be different chemistry. Phone batteries are usually Li-Ion, and power-banks are usually Li-Po. I would advise against using a different chemistry than the board was intended for (especially if you use a cheap E-bay case as the base, probably next to no protection build-in). Keep it on the safe side, don't want stuff going boom in your pocket.
Using a charger board for a different chemistry could also mean the battery wears out faster. It's not just 'oh it's under an amp so it's fine', batteries need special charging regimes - part constant voltage, part constant current etc.

Just to clear up something on the chemistry, Li-po is simply the marketing terminology used to describe the outer casing and electrolyte structure, nothing to do with the chemistry. Practically all are Cobalt, and charge to 4.2V. I've only seen different chemistry (such as lithium-iron-phosphate) in power tools and EVs. Phones use Cobalt for the superior energy density.

So long as it's a single cell, and is a lithium ion or lithium polymer chemistry (which generally charge to 4.2V), you should be ok with the circuit out of one of those portable USB batteries. The mAh probably won't matter, I doubt the USB battery circuit does energy metering.

ok, so I won't be at risk of overcharging the battery? 

Nah pretty much every lithium ion/lithium polymer battery used in consumer devices these days use lithium cobalt oxide chemistry, all charge to the same voltage with a constant-current, then constant-voltage charger. The only difference between chargers is the current they charge at, but for a USB charging circuit, it's generally limited to an amp or two, this will be safe for a cell-phone size cell.