The fact that the ability says "gain control of target creature" only means that the chosen target must be a creature at the time it is chosen and at the time the ability resolves. This is just like any other targeted spell or ability: targets have to meet the specified conditions when you choose them, and also when the spell or ability resolves.
After the ability resolves, it doesn't matter what happens to the targeted god's characteristics - whether it stops being a creature, changes color, starts being a creature again, changes power and toughness, gains or loses abilities, or so on. Nothing causes the control-changing effect to end, unless it has its own built-in time limit (for things like Act of Treason). Of course it can be overridden by another control-changing effect, i.e. player 2 could steal the god back. Here's the relevant rule:
611.2c. If a continuous effect generated by the resolution of a spell or ability modifies the characteristics or changes the controller of any objects, the set of objects it affects is determined when that continuous effect begins. After that point, the set won't change. (Note that this works differently than a continuous effect from a static ability.) A continuous effect generated by the resolution of a spell or ability that doesn't modify the characteristics or change the controller of any objects modifies the rules of the game, so it can affect objects that weren't affected when that continuous effect began. If a single continuous effect has parts that modify the characteristics or changes the controller of any objects and other parts that don't, the set of objects each part applies to is determined independently.
You're basically right about the difference between this and Mind Control, by the way. With Mind Control, the control-changing effect comes from a static ability of a permanent, as opposed to the Siren's effect which comes from the resolution of an ability. The relevant rule for Mind Control is this one:
611.3b. The effect applies at all times that the permanent generating it is on the battlefield or the object generating it is in the appropriate zone.
So in that case, the control-changing effect only lasts as long as the Mind Control is actually enchanting the thing.