What does "one another" mean in John 13:35?

No, this very is specifically about the love the disciples have for one another. The Greek is as unambiguous as the English translation you cited. (Strictly, the words are addressed to the Eleven; exegesis won't decide for you whether it applies by extension to all Christians.)

There isn't another verse that teaches directly that the love Christians must have for all people (enemies, etc.) is a witness, though it's certainly to conclude that it is.

Among other verses where Jesus teaches us to love others ...

Matthew 22:36-40(KJV)

36 Master, which is the great commandment in the law? 37 Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. 38 This is the first and great commandment. 39 And the second is like unto it, Thou shalt love thy neighbour as thyself. 40 On these two commandments hang all the law and the prophets.

Jesus was speaking not just to his disciples in this instance, but indeed to his political/religious opponents, the Pharisees.

Jesus extends this teaching of love to "our enemies."

Matthew 5:43-45(KJV)

43 Ye have heard that it hath been said, 'Thou shalt love thy neighbour, and hate thine enemy.' 44 But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you; 45 That ye may be the children of your Father which is in heaven: for he maketh his sun to rise on the evil and on the good, and sendeth rain on the just and on the unjust.

That's a pretty inclusive teaching, right there.

The "love your neighbor as your self" originates in the Old Testament, Leviticus 9: 17-18(KJV)

17 Thou shalt not hate thy brother in thine heart: thou shalt in any wise rebuke thy neighbour, and not suffer sin upon him. 18 Thou shalt not avenge, nor bear any grudge against the children of thy people, but thou shalt love thy neighbour as thyself: I am the Lord.

If you read the whole of the gospels, you will find other references to Jesus' teaching us to love our fellow man. It would be hard to conclude that "one another" only applies to the "in group" when one examines all of the verses, even though in some passages (in context) that might be reasonably inferred (as with the passage you cite).

As an aside, reading all four of the primary gospels (all of the pages, all of the words, all of the verses) is a good starting point in any understanding of the Bible. I highly recommend it, be ye Christian or not.

To understand Jesus meaning it is important to consider verses 34 & 35 together...

“I am giving you a new commandment, that you love one another; just as I have loved you, you also love one another. By this all will know that you are my disciples—if you have love among yourselves”

Jesus said he was giving a “new” commandment. New in what sense? Under the mosaic law, Gods people were were commanded to love their neighbors as themselves (Leviticus 19:18). In Jesus day the Jewish Religious leaders taught that the terms “friend” and “neighbor” applied only to Jews. Non-Jews were to be hated. Jesus explained the proper application of the law when someone asked him, “who really is my neighbor?”. Jesus responded by relating a parable that described a Samaritan who came across a man, a Jew, who had been assaulted by robbers. The lesson? Our love of neighbor should extend to people other than those of our own race, nationality, or religion. (Luke 10:25-37)

However, Jesus command to his disciples was “new” in that it was to be expressed beyond neighborly love. His followers were to show love to one another “just as He had loved them”.

Jesus loved not by a few good deeds, but by feeding people spiritually and laying down his life for others, even for those who had not yet embraced his teachings.

Jesus words further indicate that Christians would show a unique love, not only for those who are not yet believers, but especially toward those related in the faith. So different, it would stand out as an identifying mark of true worship... “By this all will know you are my disciples-if you have love among yourselves”.

Regarding the second part of your question, “are there any other scriptures that show Christians would be identified by their love for everyone?” (roughly paraphrased), the 1st thing that comes to my mind is warfare. Since Christians would not be confined to one nation, but would live globally in different nations, how would such love be displayed and identifiable in times of war? What bible verses shed light on this?

Early History reports no Christian would kill another human being. The Encyclopedia of Religion notes: “The early church fathers, including Tertullian and Origen, affirmed that Christians were constrained from taking human life, a principle that prevented them from participating in the Roman army.”

Bible verses that verify this... Micah 4:1-5 gives a prophecy about the identifying marks of true worship in the last days. One aspect of the prophecy is that gods people would lay down their swords and learn war no more.

Matt 26:52... when Peter used his sword to defend Jesus, Jesus said “return your sword to its place, for those who live by the sword will die by the sword”

2 Cor 10:3,4...”For though we walk in the flesh, we do not wage warfare according to what we are in the flesh. For the weapons of our warfare are not fleshly, but powerful by God for overturning strongly entrenched things.

1 John 3:10-16....”the children of God and the children of the Devil are evident by this fact: whoever does not practice righteousness does not originate with God, nor does the one who does not love his brother...not like Cain, who originated with the wicked one and slaughtered his brother...everyone who hates his brother is a murderer, and no murderer has everlasting life remaining in him....we are under obligation to surrender our lives for our brothers”.

1 John 4:20,21...if anyone says “I love God,” and yet is hating his brother, he is a liar. For the one who does not love his brother whom he has seen, cannot love God, whom he has not seen. And we have this commandment from him, that whoever loves God must also love his brother.

Rev 13:10...If anyone will kill with the sword, he must be killed with the sword. This is where it calls for endurance and faith on the part of the holy ones. (Certainly it requires endurance and faith to take the consequences of laying down the sword during the emotional, patriotic frenzy of warfare. Depending on the country you live in, it could mean imprisonment or death.)

For further reading on this subject....

https://wol.jw.org/en/wol/d/r1/lp-e/1101993044?q=john+13%3A34%2C+35&p=par#h=23

https://wol.jw.org/en/wol/d/r1/lp-e/2014682#h=7:0-10:220