Some clerks have claimed that, but that's akin to a vegetarian working in a butcher's shop and complaining that they have to handle meat. If the job description changes such that conflicts with your religious beliefs then you do have a choice to resign, or, ideally, be made redundant and given redundancy pay to tide you over until finding a more suitable job.
I was thinking more along the lines of how women's rights are affected, for instance. What's recently been happening to women in prisons quite exemplifies the issue.
I think the crucial difference is that nobody was disadvantaged by legalizing same-sex marriage. At least in how it was implemented in the UK, religious venues are not compelled to officiate such marriages. So the LGB folks still get to enjoy marriage equality, but the reverends and imams and suchlike don't feel disrespected.
I find it intriguing how this has become part of the conservative political zeitgeist, when it's an issue that previously was the sole preserve of left-wing radical feminists. And one they have been battling for many decades now.
Of course, the conservatives approach it from an entirely different point of view to the feminists, for whom this a fundamental women's rights issue. But it's interesting how they can all reach roughly the same conclusion from entirely different sets of principles.