Power supply to led string
Figure 1. The simple key-fob LED power circuit works on the principle of current limiting by battery internal resistance.

As mentioned in the comments, those little cells have some internal resistance which acts as the current limiter. You can confirm this by measuring the open-circuit voltage and then measure again on-load. You will see a voltage drop.

If you can measure the current as well then you can calculate the internal source resistance value from \$ R_S = \frac {V_O - V_L}{I} \$.

Your circuit seems to be using 2 x 3 V which is on the high side for a single LED but it isn't possible to figure out the wiring from your photo. The same principle of battery resistance current limiting is applicable.

Reference: Battery plus LED without resistor.
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