It has the word cross in the name, that makes it religious to idiots.
The Red Cross is a religious organisation, much like the Salvation Army. Most charities are.
Funding Sources for Red Cross
Charitable donations from the American public
Participation in the United Way and Combined Federal Campaigns
Reimbursements and grants from local, state and federal government agencies for specific projects
Don't see much religion there.
Whether they are or aren't, they are not always the best charity to donate to.