You do not need to give up as Kamil Kaczmarek suggests, and you also do not need to shoehorn a transformed variable into a logistic regression as Peter Flom suggested (although to be fair to his intent he expressed reservation about the idea). I agree with Peter that Dale C's suggestion of using a $\chi^2$ test doesn't work (due to change of variables usually entailing a different sampling distribution for the test statistic).
Instead you can develop a new model from first principles. You've noted that you have different measure scales that in some sense are getting at some shared latent construct. So go with that; you need a measurement model with two types of measurements.
You also ought to develop a causal model, which you should have done before getting into the statistics. With a causal model you can then formulate a principled statistical model, although I can't tell you exactly how to do that because your scientific domain is outside my expertise. I would suggest watching Science Before Statistics: Causal Inference to wet your appetite, and then carefully go through Causal Inference in Statistics: A Primer. Then read A Crash Course in Good and Bad Controls to get some less theoretical advice about how to use your causal model address Peter's concern about covariate selection.
Since the statistical modelling choices depend on the causal modelling choices, I cannot recommend a specific statistical model either. But here is an incomplete toy model to give you a sense of what this approach can look like.
Suppose a latent variable
$$U \sim \mathcal{N}(0,1)$$
which relates to the observed binary observations via
$$\text{logit}(p) = f(U)$$
$$X_1 \sim \text{Bernoulli}(p)$$
and relates to the Likert-type scale via
$$X_2 \sim \text{OrderedLogit}(g(U))$$
where $f$ and $g$ are suitable choices of functions. While you will likely need to include more variables (errors/latent variables/covariates), the basic point of this toy model is to explicitly show an approach where you are taking measurements of the same underlying construct $U$ which is only observed as $X_1$ or $X_2$. Keeping your causal assumptions in mind, you can follow Bayesian workflow to refine your statistical model.
What's more, if your causal assumptions allow (i.e. not MNAR), then you can use Bayesian model-driven imputation to get a probability distribution over counterfactual measurements. For example, you could obtain a probability distribution of what a given participant would have answered on the 0-5 scale had they responded on that scale instead of the binary response scale.
