The short answer is probably "yes--and you don't even need a flat prior for this argument to hold."
For example, the Maximum A Posteriori (MAP) estimate is a generalization of maximum likelihood that includes a prior, and there are frequentist approaches that are analytically equivalent to finding this value. The frequentist relabels "the prior" as a "constraint" or "penalty" on the likelihood function, and gets the same answer. So frequentists and Bayesians can both point to the same thing as being their best parameter estimate, even if the philosophies are different. Section 5 of this frequentist paper is one example where they're equivalent.
The longer answer is more like "yes, but there are often other aspects of the analysis that distinguish the two approaches. Still, even these distinctions aren't necessarily iron-clad in many cases."
For example, while Bayesians do sometimes use the MAP estimate (posterior mode) when it's convenient, they typically emphasize the posterior mean instead. On the other hand, the posterior mean also has a frequentist analogue, called the "bagged" estimate (from "bootstrap aggregating") that can be almost indistinguishable (see this pdf for an example of this argument). So that's not really a "hard" distinction either.
In practice, all this means that even when a frequentist does something that a Bayesian would consider totally illegal (or vice versa), there is often (at least in principle) an approach from the other camp that would give nearly the same anser.
The main exception is that some models are really hard to fit from a frequentist perspective, but that's more of a practical issue than a philosophical one.