A point that seems to be being missed so far is that the POWER output of the regulator is <= POWER in. If the PV panel is not able to provide the Power (= Vout x Iout) that the battery requires at a given voltage then the regulator will not be able to maintain the voltage.
Result depends on circuit used BUT the ebay sourced module that you show probably attempts to maintain set voltage regardless of Vin. It will only do so if the available power is >= the required output power. The circuit is probably somethink like fig 20 in the LM2596 data sheet that whatsisname referenced.
If the battery is provided with 13.8V it will attempt to draw a current based on its charge state at the time. The current may be relatively low if the battery is fully charged and relatively high if the battery is relatively uncharged. In all cases your available input POWER needs to be >= the power your battery is trying to take. If P_solar is < P_desired_battery then Vout_buck will fall to a level where it is able to meet the resultant load.
You need to advise PV panel max rated power in full sun. Ideally also rated voltage and current at max power point in full sun.
You also need to advise the battery capacity and battery type (flooded, absorbed glass mat, ...). Knowing actual brand and model of battery may help.
Just applying 13.8V to a battery to charge it would be VERY bad for its health if your charger had enough capability to always do this. Lead acid batteries are usually charged at constant current up to a certain voltage and then constant voltage to an end point and then some twiddly magic may be done at the end. 13.8V is the usual target endpoint for certain modes of charging. Fortunately (for most) in cases like yours the max available current from the PV panel is probably less than or not vastly more than the max you should use in CC mode. This of course depends on panel Watts and battery capacity. If this is a eg 5W panel and a 12V 7Ah battery all should be well. If it's a eg 1.2 Ah battery and a 300W panel the battery will neither live long nor prosper.
What is the 'right' voltage depends on whether you want float or deeper discharge modes of operation or more. Lead acid charging is a far from straight forward process.
Battery University - chargin lead acid batteries is worth reading.