snowflake in the context of the indieweb, is typically used to refer in a derogatory way to "snowflake APIs", APIs that are (often silo) website or service provider-specific (unique like snowflakes) rather than an open standard.
Typical sites (silos) and services create their own unique API because they think their service is special (when typically there are numerous others like it). Thus they are called "snowflake" APIs.
This uniqueness means every time you want to code for a new service, you have to spend the time to learn their particular snowflake API.
A better approach would be for new services to adopt standard APIs (social standards) so they would "just work" with existing code that supports those standards.
Lacking a standard API for an area, a service should look at and perhaps imitate what previous similar services have done / are doing.
Possible Lack of Design
It is possible that snowflake APIs aren't directly an intentional consequence of a developer wanting something beautiful and unique, but rather a side-effect of their development process (no explicit design).
In most production shops - the public facing API is rarely designed, rather a collection of changes and scaffolding that ends up being what remains (citation needed of at least some examples, if "most" is true).
Often the API is not designed so much as an artifact of the backend architecture of the site, e.g.:
Are Snowflake APIs due to lack of standards
Q: If we presume ignorance rather than malice, is it possible that the reason why there are so many snowflake APIs is due more to a lack of obvious standardized approach to creating these APIs?
Or it's possible some standards are hard to find, or it's not clear that they are a well-adopted standard.
A: Possible but unlikely. If that were the case then new services would simply mimic previous similar services, but instead, they (nearly) always just make up their own API.
Mentions of "snowflake API" or "snowflake" in reference to an API being a unique one-off, in posts, tweets, etc.
Possible origin for this derogatory use of snowflake: