Is there a market for this type of video?
http://www.jeremydoyle.com/one.mov

How would you classify this type? How would you market it?

I'm just trying to gage if this is something I could try to pursue on the side of my 9 to 5.

To add to an existing catalogue of products for a commercial operation, sure. To start as a stand-alone company/product offering... I'm not so sure, maybe.

Birthday/Anniversary rememberance videos are not really too widespread, I'd personally only bother if I had a commercial operation where I marketed to the public already and offered like products, in which case I'd just add it to the catalogue and mention it in ad copy with the rest of services.

If you do weddings/family events then I'd offer coupons at these events to help promote the service initially. It's going to be heavily reliant on word of mouth.