As a general rule, legal language is interpreted loosely with respect to singular versus plural, or male versus female (in interpreting pronouns). A clause that uses the word "tenant" can thus be construed as referring to multiple tenants, and "tenants" can also refer to a single tenant. Likewise, "he, him" refers to a third person, regardless of gender. If the intent of an agreement is that only a single person shall reside in a place, then the wording of the contract would have to say that, and you can't derive that from using "tenant" rather than "tenant or tenants".
I don't think the issue comes down to "treating y'all as one person", it comes down to whether the obligation is joint, a series of several obligations, or a joint and several obligation. You would look for expressions like "We and each of us agree...", vs. "Each of us agree...", or "We agree..." to sort that out: I assume that the language just says "Tenant agrees...", that is, there is nothing at all in the wording of the lease that resolves the matter.
Tenant (whoever that is) has an obligation to Landlord to pay rent. It doesn't matter if Tenant is 1 person or 10: you have to pay the rent. If 5 out of 10 of those people mysteriously disappear, the other 5 still have to pay a now-doubled rent per person. Each person is fully responsible for all of the lease obligations, and if you are the only reliable person in a lease with 10 parties, you could get stuck with the entire obligation.
If Tenant needs to go away for some reason, Tenant can normally negotiate with another person to assume their obligations, so Tenant would come up with an arrangement with a new person, and the new person would have an obligation to (old) Tenant – this is basically a private arrangement that doesn't involve the Landlord. However: it is pretty standard that landlords get a say in letting in new tenants, and you have a clause in your agreement that says that.
There are two ways for the old tenant to "go away". One is to completely terminate the old agreement, and the landlord signs a new lease with the new person: the old tenant is completely free of any subsequent obligations, and if the new tenant fails to pay rent, the landlord has to go after the new tenant. The other way is by assigning his obligation (as described above): the agreement is between the old tenant and the new tenant (with the landlord's consent).
The question now is, what is the meaning of the clause "the assignee shall sign a separate written agreement with Landlord and Tenant"? (Earlier, I missed the significance of "Landlord and Tenant"). The core question is whether the new arrangement is a novation, or is it an assignment? A novation requires agreement between all parties, and that is what seems to be implied here. California landlord law then tells you that this "makes the new tenant (rather than the original tenant) solely responsible to the landlord". In contrast, "Like a sublease, an assignment is a contract between the original tenant and the new tenant (not the landlord)". Since this involves the landlord, the conclusion is inescapable that this is not actually an assignment (despite the use of the word "assignee").
All of the parties to the agreement would have to agree to these new terms, if in fact there is an agreement that substitutes D for C in this agreement with the landlord (a notation). If C remains on the hook and this is just a personal arrangement between C and D (with Landlords consent) – which is not what the clause says – then you don't get a vote in the C-D arrangement.