The "universal" quality is that all people will be eligible for UBI. Everyone will receive the difference between what they make and what UBI guarantees.
It's sort of like the popular phrase in the U.S. - "everyone is guaranteed the same opportunity but not the same outcome". Thus, while all will have access to UBI, the payout will be determined by individual circumstances.
Regarding Finland's program, it's likely they had to choose a group of individuals who were easiest to justify paying UBI under current popular-thought. Therefore, they chose a group already receiving or eligible for unemployment/assistance and argued they were simply replacing one for the other. That is of course just an educated guess on my part.
So, yes, UBI itself is in fact universal although the initial rollout, or test, is highly compartmentalized.
Edit: As someone pointed out, I'm probably confusing UBI with GMI - Guaranteed Minimum Income.
It's starting to feel like we're discussing rules on laying a foundation while others are already fast at work on their fourth and fifth floor, i/e, what's the point of giving everyone a ladder if the roof is consistently growing out of reach?