The joke is a mild and uncomplicated one. The idea is that you pull the string and the doll kvetches (complains) about it. "Again with the x" is just an idiomatically Yiddish way of complaining.
Nothing to do with "no strings attached".
The whole point of a talking doll is that when you pull the string, it says something appropriate to the particular doll. Woody, the cowboy doll from Toy Story, would say things like "Somebody poisoned the water-hole!" or "There's a snake in ma boot!" Mattel Corporation got in hot water for manufacturing a teenaged-girl doll that would say, "Math is hard!"
So, a old-Jewish-lady doll would be expected to say, "Have a piece of fruit" or "Oy, do my kishkes [insides] hurt!" but all humor depends on partial reversal of expectations. If the doll merely complained, that wouldn't be funny because that's what you'd expect; if the old-Jewish-lady doll said "There's a snake in ma boot!", that wouldn't be funny because that just wouldn't make any sense.
To be funny, a joke must both confirm and confound the expectations: the old-Jewish-lady doll complains about your pulling its string, in perfect Yiddish idiom.
Comedy gold, Jerry!
Another powerful aspect of a joke: hyperbole. The doll should really say, "Again with the string? Oy, gevalt. Enough with the string already!"
Yoichi Oishi, no offense meant but I imagine you with a pronounced Japanese accent. If you tell this joke (which is not particularly well-known), aloud and at all well, to any Jewish American, that person will be telling his friends about you at parties for the rest of his life.
About 1985, I saw a Japanese stand-up comic on TV say (in an almost unintelligible accent), "Not many people know this but Japan have velly small army. Velly velly small. That why monsters always attack Tokyo first."
Still cracks me up.