How do I say something like "my company," "my plane," "my bus", etc. without making it sound like I own the company/plane/bus/etc?
"Our company" makes it seem like both me and the person/people I'm talking to own the company together.
It is acceptable to say something like, "My plane/bus was late" or "My company was sold" without sounding as if you own them. It would be considered unnecessary and cumbersome to say, "The plane/bus I was on was late." or "The company I work for was sold."
Typically, people will know you don't own any of those things, and if there's any question, people will ask.
Addressing the more general case, it's important to note that possessive pronouns don't necessarily imply ownership, possession (nor does the Saxon genitive 's, despite what it says in that link).
Often, it just implies some kind of relationship, connection. For reasons that aren't clear to me, teaching materials for non-native speakers frequently over-emphasize the "ownership" relationship (and make too much of the fact that we're slightly less likely to use 's after "inanimate" nouns - despite clear evidence that the car's engine, for example, is far more common than the engine of the car).
Regarding OP's specific context, I'd say it's a "non-problem". Native speakers don't simply assume My company went bankrupt implies the speaker owned the company, unless context favours that interpretation (as opposed to it just being the company the speaker used to work for).
But in the unlikely event that both interpretations are possible in a "non-interactive" (written) context, a considerate writer could easily disambiguate by using the company that I owned / that I worked for.
Possessives -- words like "my" or "our", or use of apostrophe-s -- do not necessarily indicate ownership. They just indicate a close relationship. No fluent speaker assumes that it means ownership.
If you say "my boots", yes, you probably mean that you own them.
But it's quite common to say, for example, "my country", and no one takes that to mean that you own the country. Rather, you mean the country where you live.
A slave will refer to "my master". He certainly doesn't think that he owns the master. It's rather the other way around. :-)
If you said, "My plane was late", listeners would normally understand you to mean the plane that you were riding on, not that you own the plane. Unless you had just been discussing the fact that you owned an airplane.
There are cases where it could be ambiguous. If you pointed to some clothes and said, "this is my uniform", do you mean that you own the uniform, or that the organization owns the uniform and has directed you to wear it? If you say, "my company", do you mean that you own the company or that you work for the company? Etc. Usually it will be either be clear from context or irrelevant. "I get a lot more respect when I wear my uniform": it probably doesn't matter whether you own it or not, the point is just that you wear it. If you say, "My company fired me yesterday", you almost surely don't mean "the company that I own". But, "I have decided to sell my company", you presumably own it. "My company produces widgets": without some context to make clear, that could mean you own it or it could mean you work there. It may or may not matter to the listener.
As others have pointed out, "my company" will usually be interpreted the way you intend based on context. If you don't feel that that's clear enough, then the other usual way to phrase this is
the company I work for
I regularly use this phrase, rather than the ambiguous "my company", but not always.
"The" works for transport: "the plane was late, I missed the bus".
I think it works for "company" in most circumstances. "The company sent me to Alaska". "Careful with the car seats, it belongs to the company".
Some dialects go further in this direction and use "the" where "my" would be unambiguous. There are people who will refer to "the wife", although this feels a bit antiquated and working-class. Never "the husband", perhaps "the old man" instead.
And Myles na gCopaleen had his great series of anecdotes in the Irish Times about "the brother", meaning his brother. But that usage feels very Irish-specific.
I use "our" to describe such things. Eg:
It's a simpler way to say that you are somehow related to the thing/object/whatever, but it doesn't belong to you (or you aren't the sole owner of the thing).
If speaking, rather than typing, intonation is key.
If you slightly stress "my", it will sounds like you own the company.
Both words equally stressed is ambiguous, and will tend to be interpreted by the listener depending on their perspective.
Stressing the word "company", though, is more likely to give the impressions you want.
It's not guaranteed, as there are many factors. But consider intonation and stresses when you respond.