If by "relevant parts" you mean the video IDs, I think those alone could be considered URIs, but not very good ones, as they do identify the resource, but not unambiguously. If I ask you to find me 4x_xzT5eF5Q, that could be difficult because there could be any number of things named 4x_xzT5eF5Q, and that name tells you nothing about what the thing I expect to find is.
On the other hand, the full URLs both uniquely identify one particular object. Nothing else in existence (probably) is named https://youtu.be/4x_xzT5eF5Q. That also happens to be a URL, so tells you once place you can find that object, but if you treat it as the object's name, you can imagine you might be able to find that object elsewhere under the same name (e.g. on some theoretical Youtube mirror). (Also, this object has many possible names: any URL which leads to it can be considered another name for it.)
(Of course if you want to make sure nothing else has the same name as your object, you should ideally use a UUID or hash. A hash also has the advantage that all copies of it automatically have the same name, and all variations automatically have different names, which are also unique to them.)
tl;dr a Youtube video ID, and any valid URL for that video, can be seen as one of many possible "names" for the video (and a URI is essentially a name), but these are poor URIs because there's no guarantee that a) nothing else has the same name and b) all copies of the video have these names.
