The term "Essigäther" is actually the German name for ethyl acetate i.e Essig = "vinegar" + Äther = "ether". "vinegar" becomes acetic, hence ethyl acetate becomes "acetic ether". (etymoline.com)
But, ethyl acetate is an ester. So, why is it named "acetic ether"?
According to Leopold Gmelin, ester was called "oxy-acids ether" or "ether of third type". The original German text is:
b. Ester oder sauerstoffsäure Aetherarten. Ethers du troisième genre.
Viele mineralische und organische Sauerstoffsäuren treten mit einer
Alkohol-Art unter Ausscheidung von Wasser zu neutralen flüchtigen
ätherischen Verbindungen zusammen, welche man als gepaarte
Verbindungen von Alkohol und Säuren-Wasser oder, nach der
Radicaltheorie, als Salze betrachten kann, in welchen eine Säure mit
einem Aether verbunden ist.
which translates to:
b. Ester or oxy-acid ethers. Ethers of the third type.
Many mineral and organic acids containing oxygen combine with an
alcohol upon elimination of water to [form] neutral, volatile ether
compounds, which one can view as coupled compounds of alcohol and
acid-water, or, according to the theory of radicals, as salts in which
an acid is bonded with an ether. (italics mine)
You see the name "ether" was used to define an ester but this usage has become obsolete to avoid confusion.
Other compound classes originally named ethers
Gmelin reports in his 1848 Handbook of Chemistry that French chemists use the terms ether of the first, second and third type to refer to ethers, thio- and halo-alkanes, and esters, respectively. While the modern term ether refers to compounds with a specific functional group, the term ether was historically used to refer to a variety of volatile substances.
The names ether and ester are currently used, while the name "afer" for the thio- and halo-alkanes has not caught on.
- Leopold Gmelin, Handbuch der Chemie, vol. 4: Handbuch der organischen Chemie (vol. 1) (Heidelberg, Baden (Germany): Karl Winter, 1848)