'The company I work for' cannot be shortened to 'My company'?

Your teacher is correct that there is ambiguity there, you could be referring to either the company that you own or the company that you work for.

However, you are correct that this ambiguity hardly ever arises in practice, for two reasons: first, there is only a small percentage of people who own a company, so you're not too likely to be in a situation where the "ownership" meaning is intended (or if you are, it will be apparent from context anyway); and second, the people who do own a company almost always also work at that company, so the distinction is quite probably irrelevant in most of those few cases where the potential ambiguity actually arises.

Also, as pointed out recently in an answer to a completely unrelated question on ELL, the use of genitive is not necessarily related to possession at all; so your teacher is not correct in asserting that my company would by default refer to "the company I own".

In a given context, it might be confusing or misleading to say "my company" to refer to the company that you work for. But it is mot definitely NOT wrong.

Possessives like "my" do not necessarily mean ownership. They simply indicate some sort of relationship. We use possessives in many different senses:

• my pencil: I own this pencil.
• my house: If I have a mortgage, do I really "own" the house? If I fail to pay on the mortgage, the bank can take it away.
• my dog: You might say that I own the dog, but I almost certainly do not view my relationship to my dog the same way that I view my relationship to my pencil.
• my wife / my husband: There is a certain sense of ownership, but most people do not see their husband or wife as a piece of property that they own. Unless you have a very bad marriage. Or maybe a very good one. :-)
• my friend: I certainly do not own this other person. I am just indicating that this person and I have a relationship, as distinct from being strangers, or him being your friend. As in, "Who invited Joe to the party? Is he your friend?"
• my country: I do not suppose that I own the country. I mean the country of which I am a citizen. I suppose a dictator might say "my country" in the sense of owning it.
• my master: A slave does not own his master. The master owns him.

Sure, a possessive could be ambiguous. If I say "my company", do I mean the company that I own? The company that I work for? In context, it could mean other things. Like, "I have a retirement fund with Fidelity. How about you?" "Oh, my company is Mercer." I don't either own or work for Mercer, I just have an account with them. Or a group of terrorists might have a conversation like, "I'm going to be part of the group blowing up the oil refinery today. Would you like to join us?" "Oh, sorry, my company today is XYZ Airlines." Etc. "My company" could mean all sorts of things depending on context. If it's not clear which you mean from the context, then you should add words to make it clear.

The ambiguity here is created from the lack of a shared knowledge base or context. To say "my company" in a business context, for example among co-workers, it would be understood that you mean the "company I work for" and not "the company I own". With a broader audience, i.e. one that doesn't share your coorporate knowledge base, this is a problematic expression, and not silly. Your instructor wants you to write clearly and that means readers must be able to resolve the ambiguites in some way.

Another way to say "my company" without being to verbose, might be to use a phrase like "my current employer" which allows you to personalize it as with "my company" but remove the ambiguity of possible ownership.

The main lessons here are that 1) english gives us plenty of options to express our ideas and 2) you need to always consider your audience when you select the words to express them. Also, kudos to you for attempting to be more concise in your writing. However, the challenge in using fewer words to is the extra work required to do so and not sacrifice meaning in your expession.

The reader could definitely confuse whether or not 'your' company is possessive or not. I, personally, do see it as a possessive trait since it feels more standard that someone who owns a company would use that terminology rather than coming up with a non-ambiguous alternative. It would be strange for them to say something lengthy as well: "The company that I own ..."

There are other examples of how to say "The company that I work for." You could definitely use a short one like "My workplace." These ambiguous statements are definitely based on the context of the individual and what type of reading it is, but this is exactly why these scenarios are situational.

Sidenote: Questions are a great example of how something simple can be confused. "You didn't go to the doctor today?" Most commonly those are Yes or No statements, but simply saying "Correct." clears up all the confusion.