It seems that with a load the solar panels' voltage dips enough to keep you out of trouble, but I agree with the others that you should keep an eye
on the electrolyte levels if it is a "wet" serviceable" battery.
A simple series diode in their design may prevent battery discharge, but it doesn't provide voltage regulation. Ideally there should be a basic voltage regulator
on the output of the panels to prevent the output voltage from going over 15V which could damage the battery. You may well get away without it when using these specific solar panels, but in a commercial design that could use multiple types of panels or panel designs would have one.
A simplified "matchbook" explanation of solar cell charging, if you are interested:
Solar cells with light shining on them produce an electric field within themselves which is somewhat self-regulating - with nothing wired up to the solar panels and impinging photons from the sunlight, the PN semiconductor junction in each solar cell has a maximum electric field potential as it tries to make donor electrons migrate across the junction from the N-type material to the P-type. Very, very,very little current results within the cell, as the electrons want to find an easier path without fighting the electric field. Once the output wires are connected to a load (you connect the wires of the solar cell to your battery) the electrons have another easier path to follow, and many of them do. However, this net migration of electrons in this easier path causes the electric field potential to decrease across the PN junction, which lowers the voltage that the solar cell generates. The more current that is directed through the output wires (more net electron migration) the lower the voltage of the solar cell goes. IF too much current is demanded by connecting too low of a resistance, or load, to the device, the voltage output drops to its minimum, which then can stop the output of current as the electric field isn't strong enough.
As long as the battery's effective resistance to this current (which is related to it charge level) isn't too low, then the output voltage of the arrangement of the solar cells in that panel is high enough to maintain the charging current to the battery. As the battery charges up and provides higher resistance to this electron flow, the solar panel's voltage starts to increase. But as soon as the voltage gets high enough to try to cause more charging current to flow to the battery again, as soon as the current begins to flow the solar cell's voltage goes down again - eventually an equilibrium point is reached. Once the battery's voltage equals the output voltage of the solar cell array minus the protection diode's voltage drop, the charging will effectively stop. The designers of those panels obviously chose the number and wiring arrangement of the solar cells so that they could be used in this type of application with minimal external circuitry. But, as time goes by and a number of cells become damaged/inoperable, the panel's ability to charge the battery will deteriorate, which a more sophisticated design could still accommodate and work effectively.
IF you tried connecting a dead (0V) battery to the solar cell, that would essentially create the situation described above with too low of a resistance - the solar cells try to output too much current, which causes the internal electric field to degrade, lowering the output voltage and stopping the output current. It cannot keep up with the demand, because of the characteristics of the device, and thus cannot charge the dead battery. The voltage you measured (13.8V with solar panels connected) is higher than the disconnected battery (13.4V) voltage indicating that current is flowing into the battery, charging it up.
Hopefully the above rambling makes some sense.
