I don't think it requires much explanation beyond being a 24th century Video Will, which has been a thing ever since video cameras and VCRs became widely available in the 20th.
It wouldn't be too out of character for Yar to record such a message. Her parents were killed when she was young, and her childhood was rough, and so she was probably attuned to the fact that life is short and friendships are precious. It wouldn't be surprising to find out that she routinely kept such a message up to date, which would explain why it was tailored to the people she was serving with.
It's also possible that she addressed people in her message that were not present at the memorial service, and that the computer (or a human) edited out those parts before playback.
As for whether it was common or not, that's hard to say. The only similar instance I'm aware of is a recorded message to Wesley from his father (episode "Family"), but Jack Crusher didn't expect to be dead before it was playbed back. No other Starfleet memorials that I can recall have involved playing back such a message (Spock's memorial in Wrath of Khan, and another in Enterprise), but those instances took place much earlier than TNG.