Is there any proof that the Taishanese language is related to Gan? Is it from a linguistic perspective?
The Gan-Hakka hypothesis is most famously put forward by Sagart (2002), based on certain unique shared innovations in both Southern Gan varieties and Hakka:

屋下 as the usual word for &quot;house&quot;, compared to Cantonese 屋企.
Use of 倈/孻 (beginning with /l/) as the usual word for &quot;son&quot;.
Similar words for &quot;here&quot; and &quot;there&quot; between 梅縣 (Meixian) Hakka and 永修 (Yongxiu) Gan.
A word for &quot;you&quot; that begins with /h~x/, has a nucleus /e~ɛ~æ/, and ends in a nasal, instead of using those that are related to 爾 ~ 你.

However, one notices that these are primarily lexical in nature, plus these are restricted to various features of Gan quite far from Nanchang 南昌, the most studied variety. As far as I know, none of these &quot;unique&quot; shared innovations are common to 四邑 Siyi/Sze-yap/Seiyap varieties.
There are nonetheless plenty of 四邑 Siyi features that mean it is definitely very divergent from the other Yue varieties, as described on that blog post; this is acknowledged among those who do hold to classifying Siyi as part of Yue.
I would also cite the following tone merger:

the merger of Middle Chinese 陰平 (yīnpíng) and 陰去 (yīnqù) tones, which in Taishanese results in a mid-level tone (33).

However, this is very different to the Hakka pattern, where 陰上 and 陽上 merge with each other, and 陰去 陽去 merge with each other (although there is a good deal of 濁上歸去, as is true in Mandarin and Cantonese). Hence this is actually specific to specific parts of Siyi, and according to the seminal work on Taishanese phonology, Cheng (1973), it may be part of a proper tonal chain shift.
The way 濁上歸去 interacts with the above tone merger also creates effects which are very reminiscent of certain key defining features of Hakka, as well as a Southern Gan variety spoken in Nanxiong 南雄, Guangdong, for example in the pronunciations of 耳 and 兩.
The absence of vowel length in Siyi is an interesting one, as Siyi like all other Yue varieties underwent the 陰入聲 split, conditioned by the 內外轉 nèi-wài zhuǎn distinction of Middle Chinese. It is true that it does also manifest as vowel length as well as tone split in Yuehai/Guangfu Cantonese, whereas in Siyi Taishanese it only resulted in a tone split. But this is characteristic of a Yue variety; indeed, for some it is the defining characteristic, according to Huang (2009).
Siyi also has a 陽入聲 split as well. The condition for this 陽入聲 split was &quot;unclear&quot; in the 1973 study.
The consonant shifts, which are iconic traits of Siyi varieties for Cantonese speakers, are actually common to lots of varieties of Yue, Gan and Min. Though the blog post does identify a common cause for the /s/ &gt; /ɬ/ shift, I'd want to see slightly more specific distribution maps first. Even for those who see Siyi / Ng-yap as being classified independently from all of Yue, Hakka and Gan (Orlandi (2019); also cf. Pinghua), the consonant shift is not diagnostic.
Also, one cannot ignore the considerable internal diversity of Siyi, summarised by Tan (2017). Its internal diversity, with very low mutual intelligibility across the counties comprising the Siyi region, just means that there is a lot more data now, and a lot more work to be done! Like much in the field of Sinitic linguistics, a lot of the classification done in the mid-late 20th century is being re-analysed in the 21st.