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There are much debate of when two dialects become languages. Apart from non-linguistic reasons like politics, quality of understanding is most often used to decide whether two dialects are effectively two languages.
However, is there anyone (academics preferably) stating something like a percentage of words being the same? And if so, what percentage are they proposing?
There are many measures of lexical similarity or linguistic distances but neither can tell you whether something is a dialect or a language outside a very constrained context. It is easy to come up with a measure for a particular purpose such as determining historical developments or automatically recognizing different languages in a corpus.
But what determines a language is as much a social and political matter as it is a question of linguistics. Even mutual intelligibility is not a solution because whether speakers of different languages can understand each other depends not just on pure lexical closeness but also on mutual exposure (which can often be assymetrical). Also, this mutual intelligibility may only be limited to a small subset of the language or only to certain subpopulations.
Compare these two sentences saying 'How do you like it here?' in Czech and Slovak:
They only share one lexeme completely and one with some level of similarity (plus 'tu' is also used in Czech), yet, any Czech of my generation would not even blink twice over the Slovak version. Yet, a small Czech child thought it was English when meeting a Slovak linguist on a holiday abroad. Czech and Slovak are very close (so close that there were even attempts to call them Czechoslovak), yet young Czech speakers are exposed to Slovak less and less and find it difficult to understand Slovak without some considerable effort. Slovaks tend to have more exposure to Czech and therefore, tend to understand Czech better. Things become even more difficult when you look at Czech and Slovak dialects because some of them are closer to each other than to their respective standards.
This is just one small illustration of how difficult it is to make a categorical statement over whether something is a dialect or a language unless you also state what purpose the distinction is serving. But sometimes one purpose is more important than another. For instance, product labels are always bilingual in Czech and Slovak, yet, in all cases I can recall they have a lexical similarity between 90%-100% (at a guess much more than a typical utterance). Nevertheless, it is considered important to keep both languages separate for a variety of reasons that transcend questions of mutual intelligibility.
So you need to be explicit about why you want to know this and what you're planning to do with the information. But if the question is just a matter of general interest then no, there's no one percentage of similar words that will suffice to labels something a language or a dialect.
