Direct Answer to Your Question (With an Explanation)
TL;DR: yes he can.
More detailed answer
You have three (maybe four) choices:
In a more detailed discussion than during character creation, try to find out the in-world reason for this restriction against polearms for a beginning Barbarian, or for Barbarians at all.
After that discussion you can:
Choose to play a Barbarian with that restriction
Choose to play a different class
Choose not to play
Choose to play a Barbarian with a temporary restriction in hopes
that once your character has increased in level and been exposed to
polearms, you can justify his learning how to use one.
- I added this last option due to a possible reason1 behind this DM's ruling. As you all play more together, you may be able to work a deal with your DM that, with some experience in the wider world, your Barbarian can learn how to use a polearm. (p. 187 PHB Training. p. 31 DMG Training. Learning a new weapon proficiency is well within the scope of a learned feat or a skill proficiency).
... one of my friends has chosen to play the DM for us because he has the most knowledge of the game.
From this we infer that he has played D&D before. What editions has he played before? DM restrictions on Barbarian starting weapons came with the introduction of the Barbarian class1 in 1985. If your DM's experience is with older editions of the game (1st ed. AD&D?) that might explain the DM ruling.
He's been helping each of us with making our characters and I've chosen to play a barbarian, but when I said I wanted to use a glaive as my martial weapon choice he tells me I can't because it's "not a barbarian like weapon".
The DM can do this, though it usually helps to explain why.
- He may have a deeper reason, and your discussion hasn't gone far enough to make sense to you.
- It could be arbitrary.
- It could be due to him not liking the Polearm Mastery Feat.
- It could be other reasons .... try to find out why by having a friendly discussion.
When I tried to make pick another polearm he says no to that as well, and when I tell him the use of polearms is in my tribal background he then declares "I'm the DM and I say no."
Consider this: your DM was helping more than one person set up a character. It might be that your DM gave you the short answer first since you aren't the only character he was helping. If character creation was happening with all of you at the same time, the "not a Barbarian weapon" with little elaboration makes some sense.
If character creation was one-on-one, the lack of further explanation makes less sense(to me).
Is this within his power to do?
Can he veto my character's weapon choice just because he doesn't like it?
Yes and Yes. Hopefully he has a better reason than "he doesn't like it" ... but if that's his reason, yes he can do that. You now go back to the choices I outlined at the beginning of this answer, with number 4 likely gone.
1 Where might this restriction be coming from?
When the Barbarian was introduced in the First Edition of AD&D, in the 1985 supplement Unearthed Arcana, the weapons allowed for a Barbarian were "any" (p.13, 1e UA) but there were limitations for a Barbarian's starting weapon choices based on the DM's ruling ... and where in the game world the Barbarian came from. (see below, extracted from 1st edition UA)
Allowing further weapons as experience levels were earned reflected the Barbarian getting out of his native area and learning new stuff ... like new weapons different from those used in his homeland. Some examples:
(citations are from Unearthed Arcana, 1e AD&D, TSR, 1985, pages 18 and 20).
p. 18 *The initial number of weapons the barbarian uses must include the hand axe, knife, and spear. Additional weapons based upon the barbarian’s native area may be chosen by the DM.* (the p. 26 proficiency table allowed 6 total proficient starting weapons, with an addition per two levels ... more complicated than 5e's proficiencies).
p. 20.
Native territory: Many of a Barbarian’s abilities depend on the native territory of the character. It is mandatory that Barbarian characters come from some out-of-the-way barbaric state or area within the campaign. Typically they are cavemen, dervishes, nomads, or tribesmen. Only such uncivilized backgrounds can generate the necessary surroundings to produce individuals of the stock from which Barbarian fighters would be drawn.
Within the WORLD OF GREYHAWK” Fantasy Game Setting, for example, there are several areas that could spawn the Barbarian subclass. The lands of the Frost, Ice, and Snow Barbarians, as well as the Hold of Stonefist, would be the homeland of Barbarians of the Scandinavian/Slavic mold. These characters would employ broad swords and short bows in addition to the required initial weapons. (snip)
Barbarians from the Rovers of the Barrens, Tiger, and Wolf Nomads would be excellent horsemen. (snip) Their {Rovers} main weapons would be the club, javelin, and lasso or short bow.
Other Nomads from this group would be most efficient at long distance signaling, and skilled in the use of the lance, scimitar, and composite short bow.
Savages of the Amedio Jungle or Hepmonaland (snip) In the Amedio Jungle, the preferred weapons would be club, blowgun or shortbow, and dart or javelin. In Hepmonaland, the typical weapons would be atlatl and javelin, club, and short sword.
Using the above as examples, the DM can tailor his barbarians to fit his campaign. Not only does the native area determine initial weapons known, but it also serves as a base of judgment for the use of secondary abilities. These abilities are severely limited outside the native territory of the barbarian, until the character becomes more effective with his or her abilities by gaining familiarity with the new area.
In Second Edition AD&D, the Barbarian Kit in the Complete Fighter book had different restrictions on starting armor and weapons, but could, once embarked upon an adventuring career, use any weapon and any armor when new proficiencies became available as levels increased.
If the above is related to why your DM is voting no to polearms, then you have an opening for introducing the polearm later as your Barbarian becomes more worldly and more experienced at later levels.
It's worth a try. Talk to your DM.
If it's nothing like that, then you have to decide how, and if, you'll play in his game world.