You have misunderstood the verses.
Here, talk is not on Hindu Dharma, Christian Dharma and so on and so forth, but about "Swadharma" or "one's own duties". That is, the duties, that are prescribed for a particular person to perform, as per the scriptures.
And these duties depend on the Varna and the Ashrama systems.
For, example as per Ashrama, what is prescribed for a student (the BrahmachAri) is not required to be followed by the Grihastha (the householder). An example- a student can not cohabit sexually but a householder is required to do so as prescribed.
Similarly, Varna-wise, what is a duty for a Kshatriya is not necessary the same for a Brahmin and vice versa.
Manu Smriti 2.189. At his pleasure he may eat, when invited, the food
of one man at (a rite) in honour of the gods, observing (however the
conditions on his vow, or at a (funeral meal) in honor of the manes,
behaving (however) like a hermit.
Manu Smriti 2.190. This duty is prescribed by the wise for a
Brahmana only; but no such duty is ordained for a Kshatriya and a
So, prescribed duties or Swadharmas vary with the Varnas (as well as the Ashramas).
For, a Kshatriya (which was what Arjuna was), to fight for protecting the subjects, in times of need, is the highest duty.
Manu Smriti 7.144. The highest duty of a Kshatriya is to protect his
subjects, for the king who enjoys the rewards, just mentioned, is
bound to (discharge that) duty
So, he can not really shy away from fighting, his Swadhrama, by citing examples of duties that belong to others. This is the main point here.
And, the person, who without performing the the duties meant for his Varna or Ashrama, tries to follow those of others, is condemned as follows:
Varam swadharmo viguno na pArakyah swanushtithah |
jivan hi saddhyah patati jAtitah ||
Manu Smriti 10.97. It is better (to discharge) one’s own (appointed) duty incompletely than to perform completely that of another; for he who
lives according to the law of another (caste) is instantly excluded
from his own.
So, a person who tries to follow the duties that are prescribed for other Varna people, by forsaking his own, loses his own Varna (or caste). This is what says the above verse.
So, in short, the word "Dharma" can mean many things like: Religion, duty, nature or tendency etc etc. One just needs to check in what way it's used here or in a particular context.
In Vedas, there is a Mantra that is mentioning the word to mean "religion" (i have used the Mantra in other answers of mine).
When they say Putra Dharma (Dharma of a son), Patni Dharma ( ..of the wife), it can not mean religion. Here, the meaning is clearly "duty".
Again, when they say that - the Dharma of fire is to rise up, that of water is to flow down - the word is being used to mean "nature or tendency".
In this case, Arjuna is simply being taught what the Dharma (duty) of his class (or Varna) is and also the fact that he is needed to follow it.