My opinion here is not one I have full confidence in, but I find it more convincing than the alternative (though the ideal solution would be a less-ambiguous wording in the rules).
Dealing vs. taking
Effects which say "You deal X damage to a target" and effects which say "A target takes X damage" seem to be treated as identical as far as I am able to discern. For example, the rules for Damage and Healing include this paragraph:
If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell's damage is rolled once for all creatures caught in the blast.
Note that this paragraph references a spell that "deals damage". However, if you check the descriptions of Fireball and Flame Strike, you will find that both of them say that creatures "take damage". In fact, there are many parts of the rules that wouldn't make sense otherwise, such as the very popular Warlock Invocation Agonizing Blast which lets you add to the damage Eldritch Blast "deals", but the text of Eldritch Blast says that the target "takes" damage.
Following from that, we now have the Spreading Spores feature which causes:
creature takes your Halo of Spores damage
And the Symbiotic Entity feature which triggers:
When you deal your Halo of Spores damage
As discussed, various other parts of the rules indicate that "your effect causes a creature to take damage" is the same as "you deal damage". Therefore Spreading Spores counts as "deal your Halo of Spores damage" and triggers Symbiotic Entity.
What if they intended for 1 die
In my opinion, at a similar level of importance to "what is the most precise reading of what was written" is a consideration of "what could they have chosen if they wanted a different meaning". In this case, I have a proposed alternative wording:
When you deal damage with your Halo of Spores feature, roll the damage die a second time and add it to the total.
It's a little more wordy, but less ambiguous: dealing damage with Spreading Spores is not dealing damage with Halo of Spores, even if the amount of damage is the same. I thought this sort of wording was common, but I haven't been able to find an exact match; the closest I could find is part of the War Wizard's Power Surge feature:
Once per turn when you deal damage to a creature or object with a wizard spell, you can spend one power surge to deal extra force damage to that target. The extra damage equals half your wizard level.
At the very least it's precedent for a phrasing like "when you deal damage [in a certain way], deal extra damage". Since the writers chose not to use this or any other phrasing that limits the extra die to specifically the Halo of Spores feature rather than "Halo of Spores damage", I find it reasonable that it was instead an intentional choice that Symbiotic Entity applies to other things which deal "Halo of Spores damage".