In general, I assume that if a coupon shows as available, then I can claim it. Sometimes this permits two claims per week for the same product, instead of the usual single claim. This has only failed on 1 occasion for me, when the problem was my app having old data, so I'm now more careful to force updates before buying products.
So, similarly regarding multiple stores, I'm expecting that it depends whether or not the coupons are linked, e.g. if you claim Love Corn from Sainsburys then it also disappears from the Asda coupons. However, in this case claiming Brew City from Morrisons did not cause any change in the Brew City coupons for Tesco or Sainsburys. I submitted a second claim two days ago, although it's still not yet been processed; this may or may not be due to it going for manual inspection because of an unclear receipt. I can't immediately recall previous examples that were successful, but neither can I remember any directly comparable opportunities.
Update: Just rechecked and seen that my second claim has now been processed successfully!