It's meant in the sense of "shelter from the rain". Why do you think it has a sense of giving up all hope? One of the main goals when creating it was to not require the creator to move the repo away from their name.
The degree to which the word shelter has negative associations for some people is not something I would see as deal breaker territory.
I will add that there are far more uses for shelter than just homeless shelter or animal shelter. Just a few examples:
A book about a New York woman's house hunt was titled "Gimme Shelter." (Author: Mary Elizabeth Williams)
The basic essentials in life are typically listed as food, shelter and clothing.
Shelter from the storm is a well known positive expression for sanctuary during a crisis.
Shelter magazines is the umbrella term for magazines having anything to do with homes, such as home plans, bed and bath, kitchen makeovers etc.
The format "(noun) shelter" is similar to and might more readily bring to mind "animal shelter" and "homeless shelter" than a phrase in the "(verb) shelter" and "shelter _____" formats.
The mind is very pattern-happy after all :)
They sound negative when you start to look at the reason for them existing. Not because shelters are bad, but because homelessness and abuse and abandonment is bad.
From a UK perspective, shelter to me makes me think of a bus stop or an awning to get out of the rain, or to get some shade.