Multiplexer: What is the Enable pin for? | Also power saving?

In many tutorials (e.g. or it is recommended to connect the multiplexer's enable pin to ground. So the IC is "switched on". When set to high all channels are disabled.

Is the enable pin only for controlling the channels together or has it something to do with power saving? Background of this questions: I want to use four 16 channel multiplexer on an battery powered project. Atm the circuit is designed with enable to gnd. I need the multiplexer only in the daytime so does it have any influence respective power consumption to switch enable high vs to ground at night, when I do not need any channel? Or is the enable pin only an additional switch to control / disable all channels together and not a general on/off switch for the chip?

The output enable is there to prevent the chip from putting signals on the common data line so that another chip can drive the line without interference. It has nothing to do with power or on/off.
What you could do is turn off the output enable for all 4, and turn on internal pullup resistor so the input pin is not floating, which causes random level switching, and thus excess power dissipation in the '328P or equivalent uC chip.
See 5.2 of this application note. Other items may be beneficial also.

AtmelAVR4013 picoPower.pdf (92.4 KB)

As I understand this chip, most of the operating power comes externally. For example, the current being switched is whatever you put into it, and the choice of channels to switch is also external. It would only use power on its own for the internal decoder (3 -> 8) which sounds like it uses 4 uA, which isn't much. You may also find that turning enable off does not turn off this decoder. I suspect that the E pin simply is "anded" with the channel choice, so that when high, no channel is selected, however current consumption would be (virtually) unaffected.