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I'd like to play a Kenku Bard in Adventurer's League, one raised by bards since a young age to hopefully become a master of the Bardic arts through copying many other skilled bards.

One thought that also came up though: Would he need an instument?

The PHB states about bard magic:

In the worlds of D&D, words and music are not just vibrations of air, but vocalizations with power all their own. The bard is a master of song, speech, and the magic they contain. Bards say that the multiverse was spoken into existence, that the words of the gods gave it shape, and that echoes of these primordial Words of Creation still resound throughout the cosmos. The music of bards is an attempt to snatch and harness those echoes, subtly woven into their spells and powers.

So, that would mean (in the words of my DM) that the musical instrument isn't a spell focus like a wizards staff, but rather an instrument to help recreate the perfect sound to touch the magical essence needed to create fire from thin air, or whatever magic you wanted.

If so, does that mean a Kenku could use Mimicry as their instrument, assuming they heard the right sounds from that specific instrument before?

Your DM does not get to make determinations like this in Adventurers League play.

The rules are quite clear on spell components. As far as material components goes:

Casting some spells requires particular objects, specified in parentheses in the component entry. A character can use a component pouch or a spellcasting focus (found in “Equipment”) in place of the components specified for a spell. But if a cost is indicated for a component, a character must have that specific component before he or she can cast the spell.

In other words, you either need the appropriate material component or a focus item. For bard spellcasting, that focus item can be a musical instrument.

You can use a musical instrument (see the Tools section) as a spellcasting focus for your bard spells.

Wizard class features have a very similar passage:

You can use an arcane focus (see the Adventuring Gear section) as a spellcasting focus for your wizard spells.

Based on the published rules, to which DDAL cleaves very closely, your DM's personal interpretation is flat out wrong: other than what physical object they are allowed to be, a bard's spellcasting focus and a wizard's spellcasting focus function in exactly the same manner. In fact, there isn't even anything saying a bard must play the instrument to use it as a focus - they just need to have it in-hand.

As for using Mimicry... a kenku's voice is not a musical instrument. Verbal components are a different category of spell component. While there are rules for using a focus-bearing-hand to complete somatic components, it still requires the character to have the focus object.

No Bard needs an instrument

Any Bard can cast spells that don't require material components without having access to any instrument at all, assuming they can provide any verbal or somatic components required by the spells. The instrument isn't strictly needed for material components either, since it can be replaced by a component pouch. Even if a Bard chooses to use an instrument, they only need to access it --- playing the instrument for magic is not required (it'd quite disproportionately favor any one-handed instruments!).

Your GM's statement is unclear on whether they're intending to change the above. Ruling that the Bard needs no focus to perform material components because of their mimicry will have a balance impact, particularly for Valor Bards who have good aptitudes for swords and shields. It'd effectively allow them to also use material components without freeing up a hand, which is a significant boon. I would recommend against houseruling this unless you, as a group, are comfortable with potentially altering the balance of the game.