Apple, Dropbox and WhatsApp

“Apple takes on Dropbox and WhatsApp.”(BBC News)
Is it going to ‘fight against’ the two companies?
Or, is it going to ‘introduce’ the same services as the two companies are now providing?

“Apple takes on Google with own maps, better Siri”(REUTERS)

In the above sentence, ‘take on’ does not literally mean ‘to develop a particular character or appearance.’

“Apple takes on Radio”

In the above sentence, ‘take on’ does not mean ‘to fight against’, I suppose.

“iTunes Radio is to feature 200 genre-based stations and allow users to make their own custom station based on a song or artist, similar to the Genius feature in iTunes.”

I think you are right. It not always means ‘to fight against’, IMHO.

I think it usually just means, to undertake, to start something, to engage…

although it could be interpreted as “battle”

 "the Related to TAKE ON
  battle, encounter, face, meet, engage"

In your first post “Apple takes on Dropbox and WhatsApp.”(BBC News), this can mean several things.

It could mean to “challenge” like in sporting events, which is like “to fight”.
It could mean to “buy the companies” which would be like to “take on” new employees, or when a boat “takes on” water.

In this case, according to this article: Apple’s iCloud Drive Takes on Dropbox and Google Drive The writer means that Apple has “entered the arena” where these other contenders are, and thus they are acting as challengers. This does not necessarily mean that it is a hostile maneuver. In the case of Apple, it is quite natural for them to want to augment their product with a feature similar to Dropbox because it makes sense. It would be a hostile maneuver if they were trying to steal market share.

However, in the case of the Reuters story you referenced, [my opinion is that] it is definitely hostile. The reason I say this is because Google maps were already allowed on the iPhone in an earlier iteration [version] of the iPhone product line. But Apple does not want Google to have that business position because they [directly AND indirectly] compete with them for profits. Therefore, Apple wants to get rid of Google maps because Google’s products are a real threat to Apple’s product line. This problem has been enhanced by Google since their introduction of the Android phone [and other products such as tablets], and their almost complete monopoly on search, and search-related advertising [Google Adwords / Google Adsense].

My comments are speculative opinion based on my personal knowledge of both companies, my personal experience with both companies, and their respective product lines and histories. I hope you find this comment useful.