From a metaphysical standpoint, we don't know if they "can" have different values. All we know is the values we observe. As legion has said, physicists tend to assume they could have been different because they have no reason to think they couldn't. It's more of a philosophical question than a scientific one. But "fine tuning" is simply the observation that if we tweak the values of those parameters in the models by very small amounts, then the models predict a physical universe which would not generate life as we know it, because for example there would be no stars or planets. So the idea that, if they can pick different values, that's an argument against fine tuning is wrong. The assumption that they can pick different values is implicit and necessary to the observation. The observation depends on the assumption. This goes back to the analogy of the 14 scrabble tiles. If there were only one way the scrabble tiles could be arranged, it would not be a meaningful coincidence to observe them in that arrangement. The meaningfulness depends in part on the fact that there could have been many values, the value observed is unlikely, and that it also suggests a hidden causal structure. Without the assumed variability of the parameters, there would be no "fine tuning".
With physics, the variability is assumed mathematically, the unlikelihood of the parameters taking on the values they have depends on the same assumption. What is an open question is whether or not the coicidence is "meaningful"
This is more or less the anthropic principle, although I think "no universe" might be slightly misleading. I think it refers to the idea that if you change certain parameters you can have immediately a big crunch where there is no inflation from the initial big bang into the "universe" as we know it, but just back into a singularity. It's still a "universe" in some sense. I think the physics gets a little fuzzy as to how to interpret the math in some of these scenarios.