Can a tag point to a video you filmed?
Yes. A tag on a print, a card or an album page can hold the address of a video, and any phone that reads it will open it. The question worth answering first is where that video is stored, because the tag will outlast most of the places you could put it.
Why this one works well
A printed photograph is already an object people pick up and hand to each other. Adding a tag to the back of it does not ask anybody to change a habit.
The tag is thin enough to sit inside an album page, under a mount, or behind a card, and it needs no power, so nothing about the object has to be maintained.
Print a line saying what happens, because a tag nobody knows about is a tag nobody taps. That line does more for the gift than the choice of chip.
Where the video lives
Somewhere with an address that will still resolve later, which is a shorter list than it appears.
A file on a phone has no address at all. A share link generated by a photo application often expires, or stops working when the file is moved into a different album.
The durable options are a page on a domain you control, or a video service where the item has a stable address of its own. Both survive; a temporary share link does not.
Test the address from a phone that is not yours and not signed in to your account. That single test catches almost every version of this going wrong.
Who can open it
Anybody holding the object. There is no way to restrict a tag, so the restriction has to live at the destination.
An unlisted address is not a private one. It is unguessable, which is enough for a family album and not enough for anything you would mind a stranger seeing.
For a gift that leaves your house, assume the address is public and choose the video accordingly.
The tag outlasts the file
A tag in an album will be readable long after the service you uploaded to has changed its terms, its addresses, or its existence.
Writing an address on your own domain and redirecting it onward is the only arrangement that survives that, and it costs nothing at the moment of writing.
The alternative is honest too: accept that this is a gift for now, not an heirloom, and do not build an album of eighty of them.
Frequently asked questions
Can the video be stored on the tag?
No. A tag holds a few hundred bytes to a few kilobytes, which is an address and not a file.
Will the tag still work in ten years?
The tag almost certainly will. Whether the address still resolves is the part to design for.
Can I stop other people opening it?
Not through the tag. Anyone holding the object can read it, so any restriction has to be at the destination.
Is an unlisted link private?
It is unguessable rather than private. That is enough for a family album and not for anything sensitive.