When something becomes a dialect vs another language seems a bit like selecting where colors change in a rainbow.
Indeed it's like trying to define the boundaries of (biology) species and variety. But still, in some cases the differences between Chinese dialects can arguably be greater than some of those within the European "languages".
To illustrate the differences between so called "dialects", "why" is "wei shen me"  in Mandarin, but "dim gai" in Cantonese; "who" is "shei" in Mandarin, but "bin gor" in Cantonese. Negatives ("no-something") usually takes a prefix of "bu" in Mandarin but "mm" in Cantonese. Their written forms are totally different too, so it's not like we have a very funny way to pronounce the same thing. I'd say it is the "basic language" that is different, while the large corpus of vocabulary is mostly shared among the "dialects". But then, vocabulary is liberally borrowed between the European languages too.
There's a few things that obscures the wide variation between the "dialects". One is that we write written Chinese in mostly the same way, due to tradition and communicative purposes. Standard modern Chinese is written in the form of modern Mandarin (i.e. Putonghua) and the norm is that "self-respecting" educated Chinese write in that standard form regardless of their spoken dialect. The second is that while our pronunciations are can be markedly different, a large part of the written script remains identical, because Chinese characters are not pronunciation based. So while there may be variations in the script in different European languages (eg. "wine" (en) => "vin" (fr) => "wein" (de)), the written text is the same in Chinese, even though there may be large pronunciation differences.
A fun thing that I like to mention is how written Japanese might in some cases be more legible to Mandarin speakers than written Cantonese. The Kanji in written Japanese has roots in classical Chinese (something comparable to Latin in Europe), and thus if the Japanese text is mostly written in Kanji (i.e. avoiding kana where possible), it's quite legible to those who understand Chinese. For example, the Kanji form of "who" in Japanese ("da-re") is written in the same character as "shei" in Mandarin. Of course, when spoken it is totally mutually illegible, but you can see how a supposedly "different" language (Chinese vs Japanese) can have more similar roots than a "dialect" within the Chinese language family.
 ("me" pronounced as in "mermaid" without the suffix)