I have the following sentence from Seneca, epistula 1, §2: "Cum placuerit fieri, toto illum pectore admitte; tam audaciter cum illo loquere quam tecum." However, I'm not sure what the "tam audaciter cum illo quam tecum"-part is? An adverbial adjunct? Likewise, this sentence: "Sed si aliquem amicum existimas cui non tantundem credis quantum tibi, vehementer erras et non satis nosti vim verae amicitiae." What is "tantundem quantum tibi"?
It's adverbial in both cases. The demonstrative adjective-modifier(?) tam modifies the adverb audaciter, which is the core of the adverbial phrase expressing how you should talk with people; and quam... is another—parallel—adverbial phrase, an adverbial subordinate clause.
Tantundem is an adverb to credis, expressing the extent of your trusting someone, and the quantum... clause is parallel.
Note that tecum and tibi are inside the subordinate quam and quantum clauses, respectively, so they have direct conexion to the main clauses.