From a Google search:
a demand or request for something considered one's due.
"the court had denied their claims to asylum"
Generally speaking, you would hear of a person in this context as "seeking" asylum.
The word "claim" would be used (in the verb sense, corresponding to the noun form above), if the person believed, usually due to the laws of the country in which they are seeking asylum, that to be granted asylum is a right, or something they are owed. In that sense, they believe they have already been offered asylum, i.e. by the country's law. They are making a claim based on the promise of the law.
With regards to the "to gain, win or achieve something" definition, I don't believe that applies here (it's in a different section to the "demand legal right" section on the Oxford Learners Dictionaries site). That definition to me, is quite abstract, but could be viewed in a similar way:
Where above I have said, "claim based on the promise of the law", someone earning a place in a team could be seen as being "based on the promise of being part of the team". It refers more to opportunity than promise, which can be referred to as a "promise", but with the implicit understanding that it is abstract as opposed to literal, i.e. there are no guarantees.
It would be similar to saying something like:
He claimed his own version of the American Dream.
It's based on the promise of opportunity, and not a literal, concrete promise, but we can still use the word "claim" to describe the fulfilment of that.