Four bits is a nibble and eight bits is a byte. Is there any formal or established nomenclature for a logical pairing of two bits?
No, there is not any formal or established nomenclature for two-bit entities. But a variety of terms do exist, some of which are listed in the Jargon File, which says:
...there have been quite a few analogical attempts to construct unambiguous terms for bit blocks of other sizes. All of these are strictly jargon, not techspeak, and not very common jargon at that.
Emphasis mine.
For a two-bit quantity, ibid. says:
2 bits: crumb, quad, quarter, tayste, tydbit, morsel