For the Summon Draconic Spirit spell, does the damage type of the breath weapon have to match that of the caster’s shared resistance?

Question

The Draconic Spirit summoned by the summon draconic spirit spell has the Breath Weapon action option (FToD, p. 21):

Breath Weapon. The dragon exhales destructive energy in a 30-foot cone. Each creature in that area must make a Dexterity saving throw against your spell save DC. A creature takes 2d6 damage of a type this dragon has resistance to (your choice) on a failed save, or half as much damage on a successful one.

It also has the Shared Resistances trait:

Shared Resistances. When you summon the dragon, choose one of its damage resistances. You have resistance to the chosen damage type until the spell ends.

Does the damage type of the Breath Weapon have to match the caster’s Shared Resistance?

Nowhere in the feature descriptions are these two linked. IMHO, the wizard chooses 1. A shared resistance type and 2. A breath weapon type. There is no RAW reason that the breath weapon is fire and the resistance is acid, for example, as long as both are chosen from the same family. My DM disagrees, and states that the wizard chooses 1 element as both the resistance and the breath weapon.

Answer 1

Each feature tells you when to choose.

Each of the feature mentioned tell you when you make a choice of damage type. For Shared Resistance, it’s when you cast the spell:

When you summon the dragon, choose one of its damage resistances.

For the Breath Weapon, you are instructed to choose a damage type after you roll damage:

A creature takes 2d6 damage of a type this dragon has resistance to (your choice)

There is nowhere in either of these feature, or anywhere else in the associated stat block and spell description, that would indicate that the “your choice” of the Breath Weapon feature is the choice you already made for Shared Resistance. We are told to make these choices at different times, by different features, and there is nothing there to tell us that a choice of one restricts your choice of the other.

This means that you can choose a different damage type each time the dragon uses its breath weapon.

Answer 2

There are lore reasons for your DM's view

I think you are technically correct that if you only read the text of the draconic spirit stat block, as written the damage resistance it shares with you, and the type of damage its breath weapon deals are not coupled, as you explain yourself and as this answer states.

But let's take a look at this:

When you cast this spell, choose a family of dragon: chromatic, gem, or metallic. The creature resembles a dragon of the chosen family, which determines certain traits in its stat block.

Now, let's for example assume you chose chromatic. Then your spirit would resemble a dragon of the chosen family. These are, according to the Monster Manual (p. 86)

The black, blue, green, red, and white dragons

If we look at these dragons, for example a black dragon is immune to acid damage, and breathes acid. A red dragon is immune to fire damage, and breathes fire damage. And so on. The spell's chromatic draconic spirit is not immune to any particular kind of damage, instead it is resistant to the types of damage from all five of them. Instead, it confers immunity to one type, and has breath weapon of one type.

The spell does not say that the form that you choose determines the immunity, or the type of breath weapon. The spirit only resembles a dragon of the chosen family. And the family chosen only determines a group of resistances to pick from. So technically by the spell text alone, you remain uncontested in your reading, and you can have a black dragon spirit that is breathing lightning and resistant to fire.

But narratively, this makes no sense. If this is a draconic spirit, and that means the spirit of a deceased or unborn dragon, and you chose a black dragon to appear, why would it confer resistance to anything but acid resistance, or have a breath weapon dealing anything but acid damage? There are no lightning breathing, fire resistant black dragons.

I think that is where your DM is coming from. You have a technical reading that really grates on the lore of what kinds of dragons exist, and what kinds of draconic spirits therefore should exist. And, unfortunately for you, you have not yet cited all the rules that are written that weigh on the matter. This also is a rule, on p. 4 of the DMG:

[A]s a referee, the DM interprets the rules and decides when to abide by them and when to change them.

So, rules as written, your DM has the right to change how this spell works, to make sense with how they interpret their game world to work. You may be able to show him that we too think the spell text would allow you to do what you want, but in the end, it is their call if they want to go with it, or not.