Is there a word for other fractions analogous to whole, half, quarter?

Not in everyday standard English, no. The fractional numbering system has become standardized. Half, third, and quarter are exceptions; otherwise the -th suffix is applied to all simple numbers from fourth onward (OED).

There may be specialized words that are used to name fractions, but they're more limited. For instance, tithe (teinds in Scotland) is usually used to denote contributing one-tenth of one's income to a church, and seems odd outside of that context (OED). Moiety is used to denote a half, usually in legal uses (OED). If there are other terms, they will likely have similar limits.