These situations are pretty different. Private employer vs. public employer, homosexuality is a protected class of people vs. alcohol drinkers who are not, etc.
Also in Kim Davis's case, there isn't really a direct religious prohibition to sign a same-sex marriage license (that I know of). It simply offends her conscious based upon her interpretation of her religion. Muslims, however, (again, as far as I know) are directly forbidden to handle an alcohol container and take it to a person for consumption. I don't know that this really matters. A deeply held conviction is deeply held whether it is correct or even makes sense, but it still is a difference in the two cases.
Either way, there isn't any reason that employers can't make reasonable accommodations. In Stanley's case, she did have an arrangement for other attendants to serve alcohol for her customers, until another attendant complained. Until that point, that seems like a reasonable arrangement.
Kim Davis also had the option of simply having another employee help out same-sex couples. That may have been considered reasonable, though, due to it being a protected class, I think it would deserve closer scrutiny. Doesn't matter, because that isn't what she did. She wasn't allowing anyone in her office to issue licenses to same-sex couples. That goes beyond being reasonable.
That would be like if Charee Stanley had been throwing the alcohol cart out the back of the plane before it took off so that no one could serve or drink alcohol.
As for suing, I don't know who Kim Davis would sue. She IS the employer. I think at one point she filed a suit against the governor, but I don't know what happened with that. If their is a lawsuit from her, I don't see how it would be an EEOC matter. It would be something else.