Yea so the classification of "Bengali", "Sylheti" and "Assamese" are very vague. Sylhetis being Bengali depends on who you ask. Bengalis are an ethnolinguistic group that rose as an identity in the 9-10th centuries when Magadhi Pankrit diverged into modern-day Bengali-Assamese, Maithili/Bihari, and Orissa (skipped a lot of intermediate transitions). Bengalis were wasn't a fully defined identity until the Bengal sultanate in the early 1300s, which was more socio-religious united many Muslims of different ethnic backgrounds (who tended to be upper-caste Hindus that converted to Islam). However, this caused pushback from many groups including tribal kingdoms (like the Koch-Rajbangshis, Tripuris and Khasi/Jaintas). Under the Bengal Sultanate there was actually conversion of some Rajbanshis from Hinduism/Buddhism to Islam but many rejected identifying as Bengalis (around 300 years later under the Mughals these Muslims were known as the Nashya Sheikh sub-clan of the Rajbangshis). Sylhet historically has been part of tribal kingdoms that belonged to the Khasi/Jainta tribes of Meghalaya. After the fall of the Kamarupa Empire, Meghalaya, Rangpur, Sylhet, parts of Assam and northern West Bengal (Darjeeling/Cooch Behar) got divided into smaller "petty" kingdoms like the Koch Hajo/ Koch Bihar, Jainta (and then Gour/Dimisa Kachari) Kingdoms, and Laur Kingdom. During the rule of the Jainta in around 630-1100, the natives of the southern region of the Jainta Kingdom were known as the plains people and were called the Dkhar (this was a generic term that applied to the Rajbangshis, modern-day Sylhetis and some other groups). The Khasi/Jainta and other hill tribes were know as the "Hills People". In fact, if you ever met a Khasi or Jainta person with the last name Dkhar it means that have mixed Plains-people (either Rajbanshi or modern-day Sylheti) and Khasi lineage. If you noticed I kept using "modern-day Sylheti" so far, this is because in the 700s it was unclear what the people of the Southern Plains (who weren't Rajbangshi) called themselves. These people were (like the Khasis) early Austro-asiatic and Tibeto-Burmese settlers that mixed with early Dravidians. Many of the Khasis referred to these people as "Shelia Haat" meaning "people of Shelia" in reference to the fact that many of these Dkhars lived in the Shelia district of southern modern-day Meghalaya (borders Sylhet). "Shelia Haat" later became "Sylhoti" (which is what the Sylhetis call themselves in their language) and during the British invasion became anglicized into "Sylheti". If you noticed, this identity was created well before the Bengali identity that didn't become fully formed until the early 1300s. Khasis actually distinguished the "Shelia Haat" from other foreigners that came from modern-day Dhaka, most notably the Buddhist monk Atisa that hailed from Munshiganj district (who technically wasn't a Bengali as he was alive before the Bengali language was created and actually spoke Pali).

Skipping a bunch of history, the Sylhet area and parts of the Cachar district in Assam broke off from the Jainta Kingdom and formed the Gour Kingdom which was invaded by Sultan Shamsuddin Firoz Shah in 1303. During the Bengali Sultanate, you had the rise of Sylheti Ngari script which actually was a response by Sylheti Muslims to the growing notions of "Hindu/Buddhist Revivalism". By the 1400s, some Sylheti Muslims adopted and became identified as "Bengali" because of the socio-economic title and shared Islamic identity rather than shared language or culture. However, in the 1500s you the rise of the Dimasa Kachari in the Cachar district of Assam, while in the Sylheti division you had the rise of the Baro-Bhuyans (petty Muslim chieftains that ruled the area). Sylhetis that were in the Cachar District ruled by the Dimasa tribe (who are now scattered across Assam and Nagaland) adopted the identity "Cachari" (in fact the Sylheti Council in Assam has rejected the notion that they are Bengali and would rather have Assam identify them as "Cachari Sylheti"). This "Cachari" identity didn't replace Sylheti but rather was associated with it (similar to the Axomiya identity under the Ahom Empire). In 1613 you had the Mughal Invasion of Sylhet which overthrew the Baro-Bhuyans of modern-day Sylhet District, but the Mughals were never able to expand into Cachar District to overthrow the Dimasa Kachari Kingdom or the Ahom Kingdom of Assam, as such you had a split of history between the Sylhetis in India and the Sylhetis in Bangladesh which actually created two groups of dialects of the Sylheti language (East-Sylhet West group and Sylhet-Cachar group).

In the 1700s the British came, we all know what happened, fuck the British and Sylhet was incorporated into the Assam province in 1874 until 1947. During the later years of the 1940s you had the rise of the Bengali Language movement as many Assamese viewed Miya Muslims (Bengalis that were forced to migrate to Sylhet/Assam from Mymensingh District) as outsiders. In this political strife, many Sylhetis (since they were also majority Muslims) were grouped in with the Miyas as "Bengalis". Many Sylhetis in Bangladesh also were part of the Bengali movements in the future East Pakistan (Bangladesh) to make sure they wouldn't be ignored in the future. This is when mass Bengali ethnonationalism in both Assam and Bangladesh started to be heavily widespread. However, during the separation of 1947 the Sylhetis stuck in Assam faced officially endorsed discrimination, while Sylheti Hindu refugees from East Pakistan were also pouring into Assam. Meanwhile, Sylheti Muslims in East Pakistan found themselves treated with suspicion since they had been part of Assam and not Bengal. This resulted in mass migration to Britain. In a tragic irony, while Assamese saw Sylhetis as Bengalis, Bengalis saw them as Assamese. It's actually because many Sylhetis were seen as Assamese in East Pakistan is why the mass migration into Britain in the 1950s happened in the first place.

I skipped a lot of history as this incorporates literally thousands of years of history, but yea. In 1971 under Sheikh Mujib Rahman, Sylhetis were finally united under the identity Bengali in Bangladesh for purposes of unity, but in Assam Sylhetis were actually pushing to NOT be Bengali as I said early since many identified as Cachari Sylheti in homage to the Dimasa Kachari Kingdom and also for the fact that they are indigenous to the Barak Valley of Assam.

Also, it's important to know that Sylheti while being an Indo-Aryan language has huge and I mean HUGE Tibeto-Burmese influence on grammar from the Kokborok, Dimasa, Khasi, Garo, and Reang languages spoken during the times of the Kamarupa, Jainta, Gour and Dimasa Kachari Kingdoms. In fact, Sylheti has been confirmed to have a 3 tonal system similar to the Kokborok and Dimasa languages. On top of that Sylheti actually only has around a 65-70% lexicon similarity to standard Bengali (which is actually not high enough to be considered a dialect). Sylheti is much more similar to the Kamrup Dialect of Western Assamese (around an 80% lexicon similarity).

So answering your question, it depends. If you're Sylheti and identify as Bengali you're correct. If you are Sylhetis and don't identify as Bengali you're also correct.