It's not in 2+1d linear EM field that there are instantons, rather in 2+1d nonlinear gauge theory. The 2+1 d free EM field is not confining, but has logarithmically rising potential, and a force that falls off as 1/r. The 1+1 d linear EM field is confining, this is the Schwinger model, but this is not a surprise as in 1d, the potential is automatically linear.
The logarithmic potential in 2+1d does mean that particles can't separate, but it is not the technical definition of a confining potential, which is that the particles should have a behavior similar to that of the 1+1d Schwinger model, that they are linked by a string with tension.
That's not a full answer, you presumably want to know what the spectrum of 2+1d QED looks like, is it all logarithmically confined positron states. I don't know the detailed spectrum of this theory, but it can be calculated, so it's a good question, but I think the question should be worded more clearly, so that it doesn't seem to be talking about instanton effects in QED. But +1, because I know what you mean.