I am one of those people who does remember everyone else's anniversaries and sends them cards. However, it wouldn't put me out to find someone else getting married on my (our
) wedding anniversary. That may be partly because I have known, in my lifetime, 7 other people who have the same birthday as me. 2 of them I have been good friends with. I have another close friend whose birthday is the day before mine. So I have been used to other people 'sharing' my celebration day for years now 
They have almost certainly just forgotten exactly which day it was, or have assumed that since the actual wedding date won't be the same, it isn't an issue. So I have to say, kindly I hope, YABU.
Plus, as has been said, their actual anniversary won't be the same, and in fact you will get a posh day out on your 1st wedding anniversary - perfect, I'd have thought! You can use the 'clash' as an excuse to go out a second time and celebrate your anniversary alone!
(On a side note, what's this thing people have of making the wedding a huge multi-part event as if it's the Olympics, with the 'legal' part in one place and a do somewhere else on a different day? I do think it would be better just to do it all in one go. Though that might make it even worse for you OP
)