The assumption with the spell and related effects is that staying on the ethereal plane takes magical power, and is not a default state for creatures not native to the plane and which have not physically traveled there via Plane Shift or similar effect. So blinking into the ethereal plane is a temporary enabled spell effect, powered by active magic, and you do not remain on the ethereal plane when the spell ends, regardless of how it ends.
Compare with the spell Ethereal Jaunt, a much higher level spell. With this spell, it is clear that the character returns to the material plane when the spell ends.
In terms of game balance, there would be serious problems if PCs had a lot of access to the ethereal plane. From there you can view the material plane, but are invisible and can move through physical barriers. Far too many low-level scenarios would be spoilt by PCs having even partially-reliable access to the plane.
As a DM, you could of course intervene, due to specific circumstances ("The rift between worlds is weak within these standing stones"), if it seems like fun. But you would be wise to keep those circumstances under plot control, and not grant all the significant benefits of being ethereal on demand:
An ethereal creature is invisible, insubstantial, and capable of
moving in any direction, even up or down, albeit at half normal speed.
As an insubstantial creature, you can move through solid objects,
including living creatures. An ethereal creature can see and hear on
the Material Plane, but everything looks gray and ephemeral.
Blink Dogs cannot become fully ethereal. If they could, the monster description would say so (because it is a very significant power). However, they can interact with ethereal creatures, and so can the user of a Blink spell. In some situations, this can be a useful side effect of the spell.
Note that Plane Shift is balanced by lack of precision, and also should the DM wish, by only having certain destinations allowed by restricting which spell foci are available.