Can "Sure" be used to respond to "Thanks"?

In OP's context, Sure as a response is simply a polite/vague/positive word with friendly connotations. It doesn't really mean much at all; you could compare it to replying with something like Okay, No problem, or Don't mention it.

In many other contexts, sure more emphatically conveys definitely, really, absolutely, very. But that meaning has been watered down in OP's context, being one in which no real information needs to be conveyed. This almost total loss of meaning is common with "formulaic" responses in low-key social interactions.

By way of illustrating why I don't think semantic analysis of this usage should be taken too far, I'll just say that to my mind a smile or nod "means" much the same in context as any of the verbal alternatives suggested.

It is not uncommon to hear Sure in response to a word of thanks. I use it often myself. For most, it abbreviates the expression of one or more, or all of the following sentiments:

• Not a problem
• Glad to help
• My pleasure
• Anytime
• I'd do it again
• You can count on me
• No big deal
• You deserve it [and more]