The project was completed "under" or "by" a company?

If one says that the project was completed by Microsoft, one focuses the attention of the audience on the corporation and directs it away from the human beings who did the actual work that constituted the project. That is appropriate in some contexts, such as a discussion of the intellectual-property rights or legal responsibilities that the corporation as a legal entity may have with respect to the project.

On the other hand, using the word under directs the attention of the audience to the fact that, whatever legal rights this corporation may have to the project, there is a sense in which somebody else did the actual work. One is thus likely to see such formulations in the contexts in which something has been said or is about to be said about those who did the work 'under' the corporation.

It should be noted, though, that the word under, by itself, does not make it clear what exactly is the relationship between the corporation and those who did the actual work. It is thus not a good idea to say 'The project was completed under Microsoft'; it is better to say something that makes the nature of the relevant relationship clearer: e.g. 'The project was completed under the sponsorship of Microsoft' or 'The project was completed under a contract with Microsoft'.

The meal was cooked under Bob??

Things can only be completed by an actor.

Your quoted uses of "under" are incorrect. Seeing sentences that use "under" doesn't mean they're right: business English, especially from non-native speakers, is filled with broken grammar (often from people trying to sound impressive).

"Under" is not a substitute for "by"; "by" tells you who or what was directly responsible, where "under" describes a circumstance of the action. Correct examples of "under" include:

• The project was completed under the guidance of Microsoft engineers
• The new rules were drafted under the auspices of the UN
• The children built their rockets under the supervision of the teacher
• The houses were built under the Five Year Plan of 1932

None of these are suggesting that the teacher, the Five Year Plan (an inanimate object!), the UN or Microsoft actually did the thing, only that they assisted in its completion (the actual actors are not identified in three of the sentences).

"auspices" is a word that appears near "under" in these kind of sentences. Its literal meaning of "auspice" is quite different (blessing, prophecy or omen), but when used in this construction it means "patronage, oversight or supervision"