Cause the leds have a minimum working voltage (also known as dropout voltage), other that a current, and if you have not enough voltage, your leds does not lit.
UV leds, usually have a dropout voltage around 3V, and if you power them with less than this value, there's no current flowing in them, and no light ..... and, ofcourse, when you connect more than one led in serie, the dropout voltages will sum ..... so, 2 of them in serie, requires at least 6V, and 4 of them in serie, requires at least 12V, for work ..... then all the strips are connected in parallel with the wires coming from the batteries (and here, now, is the currents that sums, and it's a different thing)
With 4 alkaline batteries, that are usually 1.5 / 1.65 V, you can get around 6.4 / 6.6V, so you cannot use more than 2 leds for each strip ..... and also, when they discharge, the leds becomes low luminosity more quickly ..... with 4 Li-Ion cells, instead, that go from 2.7V (discharged) to 4.2V (fully charged), wih a medium value of 3.7V, you can obtain a medium value of 14.8V, with a maximum of 16.8V when they are fully charged, and a minimum of 10.8V when they are fully discharged, before the protection circuit turn them off (always better to use "protected" cells) ..... this give you the possibility to connect til 4 leds in serie, for each strip (4 of them needs at least 12V for work), and also they will work for a longer period, when the cells are discharging .....
The current, instead, is a different thing ..... in poor words, you can set the resistor values in the way that each one of the 4 leds strips adsorb around 15mA each (using Li-Ion cells and 4 leds for each strip, the resistors value is around 200 / 220 ohm (220 ohm resistors gives a little less luminosity, but it's a standard value and easy to find in electronic parts shops) and can be used 1/4W or also 1/8W (smaller) resistors, without problems ..... now, 100 leds divided in strips of 4 leds, are 25 strips, each one need 15mA, so connecting all them in parallel, they draw from the battery pack 15 x 25 = 375mA ..... the batteries can power all them, cause this is the current, not the voltage, that they have to give (a fully charged 4.2V battery CANNOT, for no reasons, give you more than these 4.2V. no matter how much you torture it
p), but it can give you different currents for different times, cause this is its energy capacity ..... again in poor words, if a battery is rated, as the 14500 cells, for 4.2V - 900ma/h, this means that it can give you a maximum of 4.2V (usually, a medium value of 3.7V), AND a CURRENT of 900mA for an hour, OR 1800mA for half hour, OR 450mA for two hours, and so on (it depend from the total current that your load require) ..... so, with a theorical capacity of 900mA/h, the pack can power your leds for 900/375 = 2.4 hours (i said theorical, cause the real capacity is always a bit less, so you can expect a pair of hours of autonomy) ..... and 2 battery packs in PARALLEL, will double this value.
A different way for obtain the double of the duration is to use the two battery packs in serie, and then make strips of 8 leds, instead of 4 ..... this way, you end with 12 or 13 strips (96 or 104 leds in total), each strip adsorb 15mA. and all them in parallel adsorb 180 / 195mA, so the duration of the cells is still doubled ..... only, you need to decide yourself if for you is more easy to wire and sew strips of 4 leds or strips of 8 leds, when you are deciding where to place them on the fabric ..... maybe, if the strips of 8 leds are more easy for you to fit and sew on the fabric, this will end being the more easy way for you, for build it ..... by the way, with strips of 8 leds in serie, the resistors are better of 270 ohm, also this is a standard value, but you can still use 220 ohm (a bit more light from the leds, and a bit less duration time of the batteries, but still around 4 hours, if fully charged)
Ofcourse, for use battery packs in parallel or in serie, there's a basical rule to respect ..... batteries MUST be used always in the same group, and always charged in the same way ..... you cannot mix full charged and half charged batteries, or use part of them for something and then mix them with others ..... if you use 2 packs in serie or in parallel, the order of the batteries is ininfluent, but the batteries still needs to be fully charged all at the same level, before you use them.
EDIT: ofcourse, i'm giving these values assuming that the seller is selling you standard UV leds with around 3V of dropout ..... if they are different, a little variation in the values will solve the problem