I think the reason it's confusing is because a certain key phrase, that is definitely and clearly implied, is omitted for brevity:
"That's my rubber and [that's my] ruler."
The sentence applies the singular linking verb "is" in two separate cases using two separate singular subject complements.
The plural of "that" is "those". It would have been more correct to say:
Those are my rubber and ruler.
However, I actually think that sounds wrong though I am certain it is correct.
It is probably more common to say it like this:
Those are mine. (you have to point to make it clear)
That pencil and that eraser are mine.
Those two are mine.
"That's my pencil and eraser" is definitely not correct but native speakers do sometimes speak that way. It sounds like you are considering the pencil and eraser to be "one" thing.
By the way, I can't speak for all English speakers, but we do not call it a "rubber" in America (I would not know what you were talking about if not for context). We call it an "eraser".