When I was diagnosed coeliac my shopping bill shot up, and it's not just the obvious headline stuff like GF pasta and bread being more expensive.
Firstly, there were an awful lot of ordinary things that, before coeliac disease, I wouldn't have realised I'd have to buy the usually much pricier Free From version of. Pretty much every breakfast cereal, oatcakes, porridge oats, condiments, lots of stuff.
Secondly, my shopping choices across the price range are vastly reduced. Maybe, for example, there's bulk bags of dry ingredients super cheap — pulses, rice, dried fruit, nut/grain flours, whatever — but when I scrutinise the back it's got a "may contain" warning, or a gluteny minor ingredient (like suet being coated in wheat flour). So I don't risk it, and buy a pricier little brand-name bag with no warnings instead. Sometimes, the brand that happens to be cheapest or the bulk bag also happens to be one without gluten in it, but sometimes not. Repeat this for almost every item in the supermarket that's not an actual vegetable or lump of meat. I have to buy whichever one won't make me ill, not the one that's best value.
Thirdly, dropping down a range can be more difficult. Sometimes, you get one option and that's it e.g. there's one brand of oatcakes — buy it or don't. And when there are options the quality drop in GF foods can be much greater IMO —for example, going from ordinary branded white sliced bread to own-brand wouldn't have been a massive difference to me, but going from branded GF white sliced bread to (some) own-brand GF white sliced can get you something with a texture like a 70s sanitary towel.
And very often, when you see a product that's been made gluten-free, like a muesli bar or something, it's also bundled in with other premium-priced features like plant-based, paleo, nut-free, or luxury options (like how premium sausages tend to be gluten free), pushing the price up.
I also end up making more of my own stuff at home due to lack of (or inadequacy of) commercial GF versions, which usually isn't cheaper than buying ready-made things, especially since all the above applies to all the ingredients. GF flours that don't have "may contain" warnings are usually expensive, then there's things like the cheapie baking powder being bulked out with wheat flour. And the person upthread who recommended baking bread with almond flour, I'm just 🤯 at the idea that you could save money doing that.
Reduced sections are harder to take advantage of, too, unless it's plain meat and veg, or the occasional glorious packet of ultra-cheap GF treats 
Then, for me at least, there's a psychological aspect — the extra time and mental energy that shopping takes me (especially since I have other medical dietary needs) means that I have fewer mental resources left for focusing on money-saving. And after having to put back lots of things because they're not safe for me, I'm more likely to say, "Oh sod it, I'll treat myself" when I see something I like that I can safely eat.
I guess if I really had to save money, I would, though. I know all the things I've mentioned sound like wimpy excuses, but put together they've had a real impact on my shopping bill… Even so, though, it's definitely possible to spend less than I do. I could avoid GF bread and breakfast cereals, eat more rice and potatoes. DP could go back to gluteny versions of things like pasta and bread (though our setup means the cross-contamination risk is hard to manage). And even though the cheapest GF porridge oats in Tesco are more than four times the price of the cheapest ordinary oats, porridge is still a pretty affordable meal.
But it's a lot trickier to keep costs down than I'd ever have realised before.