I would not call it fake. I would call it dishonestly manufactured and sold. the problem is a proper rechargeable 3.6 volt battery would be more expensive to buy single piece than i assume the whole thing cost. At least it was a genuine solar cell. Use it for some other project or something.
In case there is a charging controller in there, I guess if one wanted to make a real solar flashlight from it, you could add a LIR2032 battery (or different size, whatever the current battery size is). From Ali, 20pcs cost 8.54, so 0.427 per piece. For manufacturers at multiple thousands of pieces, it would be a lot cheaper, but those lamps probably have to use the cheapest possible parts.
Getting a single unit IS more expensive, so possibly one of the cheapest options is currently 4pcs / $3.2, $0.8 per piece. These also are not tabbed, and would need direct soldering and some kind of tape or non conductive glue used to provide proper insulation. Soldering batteries is not impossible, as Dr Carlson (and surely others) have demonstrated. With these cr/lir types, adding insulation before should help prevent shorts during the process.
But, most likely there is no charging circuitry, in which case there's no way to really make this properly. I doubt there's enough space in the lamp case to add anything.