Interesting question. You make me aware that in English we use "tangled" very literally. There are two main questions in the OP so I will address them separately. I will address the question re the two tangled light strings separately in the second part.
PART 1: LIGHT STRING & TEDDY BEAR
There are two different foci for this "tangled string" incident, and each focus has its own correct way of speaking.
Focus on The Light String
Do we say "The light string is tangled IN or ON the Teddy bear"?
We say neither. We say: The light string is tangled AROUND the Teddy bear." I would say "around" because the light string is literally wrapped around the toy, it is not sitting on top in a tangled mess. Nor is it inside of the Teddy Bear.
If you look at the picture of the steering wheel, you see that the rope passes through the middle of the wheel and around a spoke. It is literally tangled in the wheel. Likewise with fingers "tangled in hair," the fingers are literally inside the hair. In the picture of the Teddy Bear, the tangles are all on the outside, i.e. nobody cut open the fabris and stuffed the light string inside of the Teddy Bear. Thus, it is incorrect to say "The light string is tangled on..." or "The light string is tangled in..."
Why can't we say that the string is tangled on the Teddy Bear the same way we put on clothes?
That would be like saying: The baggy old suit is crumpled on him.
The suit might be crumpled (wrinkled) and baggy and old, but it doesn't "crumple on him." "Crumple" is not an action a suit can perform.
The man put his suit on himself, i.e. he pulled it onto his own skin. Things the suit does to him:
- The baggy old suit is an embarrassment on such a handsome young man.
- The baggy old suit hangs like a flour sack on his slim frame.
The suit is an embarrassment on [him]. (Maybe his mother is embarrassed to see her handsome young son in such an ill-fitting suit but it's the best she could afford.)
The suit hangs on his slim frame.
Likewise, the light string may be tangled and it may be on the Teddy Bear like a crumpled old suit, but "tangle" is not an action a light string can perform. The string "is tangled"; it did not tangle itself. This is exactly like the crumpled old suit; the suit "is crumpled" but it did not crumple itself.
That is why I think it is incorrect to say "The string is tangled on the Teddy Bear."
Focus on Teddy Bear
Could we say:
The Teddy Bear got tangled in the light string?
I think that makes sense because the focus is on the Teddy Bear and the Teddy Bear is inside the tangles of the light string. That is different from saying "The light string got tangled in the Teddy Bear." This is like someone cut open the Teddy Bear, tangled up the light string, and stuffed it inside the Teddy Bear. That's not what happened; the light string is tangled around the outside of the Teddy Bear.
Tangled Teddy Bear
With the focus on the Teddy Bear, we might also say:
The Teddy Bear is tangled up in a light string.
We say "tangled up" the same as we talk about being "wrapped up" in warm clothing to go outside on a cold day. It's just a way of speaking and does not mean the direction of "higher up" or "up toward the ceiling." We say "in" because he is literally inside the coils and tangles of the light string.
PART 2: TWO TANGLED LIGHT STRINGS
Regarding the two light strings. First, you don't use an article before the word "string" when you name the strings. Proper nouns (names of people and places) are not preceded by an article. Thus, we don't use an article, either, when talking about objects with names, e.g. String A and String B.
There are a number of ways this can be correctly stated:
- String A got tangled in String B.
- String A got tangled with String B.
- String A got tangled in with String B.
I think the meaning is the same for all of them. English has many correct ways to say some things. I'm not sure why these words are used but they are. However, if we're putting away Christmas things we don't normally use such formal language.
A more casual way to talk about the two tangled strings is this:
The two light strings are tangled together.
We say they are "tangled together" because they are "together in one mess," not apart or separate. We could also say:
a. The two light strings are tangled up.
b. The two light strings are in one tangled mess.
Thus, you can see that there are a number of correct words to use with "tangled," but normally they are quite literal. This is especially clear with the Teddy Bear example in which we say he is "tangled up in" the strings.