Are the two of you using the same answer to the question "Does a X% chance of 2 events (in given interval) contribute X% or 2X% to the chance of seeing one event?"? That is, do you mean [time for a 50% chance of seeing ≥1 decay] or [time to see, on average, 0.5 decays].

In my case I'm talking about the (average) time until the first decay.

If you put two atoms A and B in an box and wait 1/2 of the half life, then the probability that the atom A survives is 1/sqrt(2). The probability that the atom B survives is also 1/sqrt(2). So (assuming independence) the probability that both survive is exactly 1/2. So you are right!

(I'd prefer to say that it's a property of the exponentially distribution, not the Poisson process.)

Poisson distribution is the limit of having infinite particles around.

It is, although the only difference is that the rate of this particular process changes after the first decay. Which makes it more of a continuous time Markov process, but the general idea is similar.