It's just the internal battery of the DC. If you haven't used the machine for some time it goes flat and forgets the date. If you use it more frequently and have it on for a while, the date is remembered at least until you next don't use it for some time. In some instances the internal battery may be dead so it will always forget the date etc. You can still use the machine fine though.
I hardly use my DC now so I'm used to it forgetting the date.
Putting batteries in the VMU is probably a waste too unless you take it out and about and play some of the VMU games with it. They go through them in no time too. VMUs work fine as memory cards without batteries. The long beep on startup from the VMU means that there's either no battery in, or that it's flat.
What you've described sounds normal to me.
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