Remember "isolate" doesn't mean "shown to be unrelated to any other language". It means "not shown to be related to any other language" (in a sufficiently convincing manner to establish a consensus).
Basically, Korean is considered a language isolate because modern linguists expect relatedness to be demonstrated by showing there is a significant amount of vocabulary (possibly including morphemes, not just entire words) in each language that can be derived from a common source via a regular set of sound changes for each language.
Figuring out these sound changes, and determining the form of the common vocabulary, is the process of reconstructing a Proto-language.
For Japanese and Korean, despite the superficial similarities, nobody has constructed a convincing proto-language. Wikipedia mentions some of the criticisms of proposed Altaic sound correspondences between languages.
There are good reasons for requiring evidence of common vocabulary. Areal effects have a large influence on the pronunciation of languages, and they can also affect grammar.
For example, the Basque language doesn't have any common ancestry with Spanish, as far as we can tell, and if it did they certainly must have diverged far earlier than Spanish and Romanian. However, Spanish pronunciation shares some features with Basque that it doesn't share with Romanian:
- Spanish and some dialects of Basque have 5-vowel systems /a e i o u/. (Some Basque varieties have vowel systems more similar to French, with /y/ and nasal vowels. This is another example of areal effects.) Romanian has 7 main vowels /a e i o u ə ɨ/.
- Spanish and Basque contrast a trilled rhotic /r/ and a tapped/flapped rhotic /ɾ/. Romanian has a single rhotic phoneme /r/.
- Basque and some varieties of Spanish have the retracted sibilant [s̺].
An example of an areal grammar feature is the "have"-perfect in Western European languages. The verbs involved (reflexes of Germanic *habjaną and Latin habeo) are not cognates, so it's impossible for the structure to be inherited from a common ancestor.
As Midas points out, different people have different viewpoints, and "dismissed" probably has an overly strong connotation of a final judgement. It's appropriate for Wikipedia and any other source trying to be "neutral" or conservative to classify Korean as an isolate, since the mainstream view (apparently; I'm not an expert) is that it has not yet been shown to be related to any other language group with a high degree of certainty. But different linguists have different leanings toward "splitting" languages apart or "lumping" them together. It's possible and perfectly legitimate for someone to think the languages are likely to be related even though we can't definitively demonstrate it yet.