Laypeople may not use the same blessing as a priest, so effectively the answer to your question is no, laypeople may not bless throats.
Laypeople may give personal wishes to other people, as they do when they bless their children or wish their friends and relatives blessed by God. When a layperson blesses you, no matter what props they’re using, this is exactly what you’re getting from them.
Laypeople may also bless themselves – by making the sign of the cross on themselves which I’m sure you have done many times, just like we all have. [And in fact, if I want to be blessed by a layperson, I will do it myself. I’m a layperson!]
However, you did nothing wrong in receiving what you were told would be a blessing from a layperson. There is nothing sinful in this on your part. You didn’t know, and at any rate it’s not very serious either way. The blessing of throats is a custom associated with a feast-day and that’s all.