I feel Dream Pop/Shoegaze is kind of like the child of the forgotten middle child between New Wave and Post Punk, known as Dark Wave.
New Wave, as we know it, is the eldest child of Disco and Powerpop, but raised by Powerpop's older sibling, Punk. So early New Wave bands like Blondie, the Gogos, and Talking Heads largely got their start in the punk scene, like CBGB in New York. Blondie came out with Power Pop songs like this which happened to have Destri on keyboards:
Punk had its own child, Post Punk, and there was sort of something between New Wave (and its twin sibling Synthpop*) and Punk and embraced a number of other styles, and somehow this other child existed between New Wave and Post Punk that was kind of ignored, people kept mistaking it for post-punk and/or new wave, but she was really her own person: Dark Wave. Dark Wave bands included Siouxsie and the Banshees and Cocteau Twins, who kind of jumped back and forth between New Wave and Post Punk. And they began playing music kind of like this:
And the above is an early example of what became known as Dreampop or Shoegaze.
Cocteau Twins helped build the label 4AD, which became largely synonymous with the Dreampop/Shoegaze sound, which IMO is more of in the alternative family: speaking of which, 4AD was also home to Pixies (Noise pop, a late 80s variant of power pop) and The Breeders (a grunge style band) in the 1990s; a parallel of Grunge music, who is kind of a grandchild of power pop + hardcore punk, and avant-garde + folk-rock. But I'm getting off topic, by 1990, we had one of my favourite Shoegaze bands, Lush:
So, it's definitely a cousin, but I would say it's in the powerpop/alternative rock family more than the wave/Synthpop family.
* New Wave and Synthpop look very much alike, they are twins, after all. They normally have keyboards and guitars; they have one difference, Synthpop always has keyboards while sometimes drops the guitars, and New Wave always has guitars while it sometimes drops the keyboards.
But Lush DOES have a relationship with Synthpop as Ladytron's Daniel Hunt produced Lush's comeback album.
Ladytron is a great band:
I describe myself as a little dose of toxic masculinity.