No, the type of sounds can be grouped into distinct language families and further down into language "branches". I just don't see (no pun intended) how this translates into different sign languages.

Any time you get enough deaf people in one place, who don't already have a signed language, they invent one. In times and places where signed languages have been suppressed, they've popped up all over the place at schools for the deaf. There's not much reason for any of the languages to share grammar and such with each other, since they're effectively unrelated. You wouldn't expect Japanese and Navaho to use similar grammar or vocabulary, right? Same here.

[Not Deaf, but I studied ASL and deaf culture for a few years at 2 colleges and in private lessons.]

There certainly are relationships between sign languages; A(merican)SL and French SL are arguably two dialects of the same language, or at least they were in relatively recent memory. Others are distantly related or unrelated; B(ritish)SL is absolutely unintelligible to signers of ASL and arose separately, for instance. Many of the modern global sign languages are related to one or the other of those families (and might be "dialects" as you want to call them, or might be further apart and thus merely "related languages"), but there are a few others that are not, including one—Nicaraguan Sign Language—that arose from scratch less than 40 years ago.