Current color fundus image registration approaches are limited, among other things, by the lack of labeled data, which is even more significant in the medical domain, motivating the use of unsupervised learning. Therefore, in this work, we develop a novel unsupervised descriptor learning method that does not rely on keypoint detection. This enables the resulting descriptor network to be agnostic to the keypoint detector used during the registration inference. To validate this approach, we perform an extensive and comprehensive comparison on the reference public retinal image registration dataset. Additionally, we test our method with multiple keypoint detectors of varied nature, even proposing some novel ones. Our results demonstrate that the proposed approach offers accurate registration, not incurring in any performance loss versus supervised methods. Additionally, it demonstrates accurate performance regardless of the keypoint detector used. Thus, this work represents a notable step towards leveraging unsupervised learning in the medical domain.