How does one test that the next coin toss has a % chance of landing on
Toss the coin ten times and record the results. You can use this to estimate the probability that the next toss will be heads or tails. It will just be an estimate, but it will be a meaningful estimate. Just try it—you'll find your estimate works much better than if you just picked a random percent chance for heads or tails.
Every single thing ever measured in physics is like this. Some things we have incredibly, miraculously precise and accurate measurements of, but they're still estimates. You can never measure anything with true, perfect accuracy in the physical world—you can only pretend to (that's the mathematics).
Is there a property or concept in reality where the chance of the next
single coin toss or your next dart throw hitting the bulls eye
actually means something?
You bet. Try it—toss the coin ten times or throw the dart ten times. If you're not satisfied at that point, do another ten, and you'll get a better measurement. You'll see how real it is. It may not be perfect (it can never be perfect) but it will be much better than arbitrary, and that does mean something.
EDIT: In a comment, Ludwig V pointed out that this answer is based on a discussion of multiple events, whereas the original question is based on the chance of a single case, implying that I take the supposition of a single case as somehow flawed. That's true. They also asked me to explain my reasoning, so I will.
With a question like this, I think it really helps to think in pragmatic terms—as in, what is the utility of assigning a probability to something like the outcome of a coin toss? I'd say the most obvious, fundamental motiviation for doing that is to try to predict it. Saying that a coin toss has a 50% chance of heads or tails, if true, is appealing because it tells you that you have no reason to prefer heads or tails, like if you're betting on the outcome. People make use of this idea frequently in daily life.
In line with this, when someone says that a "single coin toss" has a 50% chance to be heads or tails, they are implicitly making an educated guess about the result of the coin toss. Hopefully you agree this far. For them to truly make an educated guess, it implies that the "single coin toss" is not actually single at all—rather, implicit in their guess is that the coin toss in question has many things in common with all the other coin tosses the person has observed. People are comfortable making guesses like this because the outcome of most coin tosses in daily life does at least appear to be reliably split between heads and tails, at least for practical purposes.
All that happens when you perform the next coin toss, or throw the next dart, or what have you, is that you add another sample to the pool of all the coin tosses or dart throws you've observed in your life, which may slightly refine your overall sense of the probable outcomes of those events. To calculate the chance of the coin landing on its side in the next toss, all you need is a record of coin tosses; you can see how many times the coin landed on its side and extrapolate from there. Your guess may not be perfect—again, it can't be perfect, as we can't measure anything in the physical world with perfect accuracy—but given that coin tosses tend to resemble each other, it will probably be a better guess than if you guessed randomly. Just because the new event is unique in some sense doesn't mean it has no resemblance to other past events of its kind.
Overall, I think it's important to note than neither a coin toss nor a dart throw can ever truly be a "single event" with no relevance to other coin tosses or dart throws. Obviously each one is unique in some sense, but they must have some resemblance to other similar events or there would be no way to make reliable predictions about their outcomes. If we knew that a truly unique event was going to occur, with no resemblance to anything that has ever happened that we know of, I think it's fair to say that we could not reasonably make any predictions about it, but such an event, by definition, can't even be imagined any further than that.