They probably can tell because you're not logged into the same computer at the same time doing multiple things. I once got very involved in some online gaming where I had an alt for spying purposes opened at the same time and got banned for a few days. They actually figured it out because I used my main email address as my backup email address on one of them. It's good I don't play that one anymore!

Clare wrote to you, "how do I know you're not the same person?". I would write back a short and sweet, "I promise you we are different people. Is there anything I could/should do to confirm this? Thank you for your help!"

Catch more bees with honey than with salt. (Is that the right saying?) This type of social confrontation isn't always easy but you can do it! Good luck.