Why is '-ethane' in 'methane'?
meth vs eth

[OP] Why is '-ethane' in 'methane'?

This is a coincidence.
Methyl is

ultimately from Greek methy &quot;wine&quot; + hylē &quot;wood.

Source: https://www.etymonline.com.
The terminology was created by Dumas and Péligot in 1834 to distinguish wood alcohol from ethanol, and was published first in the French language:

Source: https://gallica.bnf.fr/ark:/12148/bpt6k6569005x/f15.item
Ethyl is from

Greek aithēr &quot;upper air; bright, purer air; the sky&quot; (opposed to aēr &quot;the lower air&quot;), from aithein &quot;to burn, shine,&quot; from PIE *aidh- &quot;to burn&quot;

Source: https://www.etymonline.com.
Liebig had argued in 1834 that what now is known to be $\ce{-C2H5}$ is a &quot;radical&quot;, i.e. a reoccuring group in molecules such as diethyl ether, ethanol and ethyl ethanoate, and gave it the name ethyl.

Source: https://babel.hathitrust.org/cgi/pt?id=uva.x002457887&amp;view=1up&amp;seq=30
Using the ending -yl for both the methyl and the ethyl group is a suggestion by Berzelius from 1835 (the correct atomic weight of carbon was not know yet, so the chemical formulas are off by a factor of two):



Source: https://books.google.com/books?id=1DM1AAAAcAAJ&amp;pg=PA376#v=onepage&amp;q&amp;f=false, see Wikipedia entry on Ethyl group.
methane vs ethane
That these both end in -ane is not a coincidence - all alkanes end in -ane. As stated in M. Farooq's answer, this nomenclature was introduced by A. W. Hofmann in 1865.
Acknowledgements
Maurice's answer inspired me to find the first work by Dumas that led to the methyl terminology. I have edited my answer to incorporate information from the excellent answers by M. Farooq and Jan and the insightful comments. In trying to figure out the sequence of events in the 1830s, I made use of this account by Frederick E. Ziegler.
References
Dumas, J.-B, Mémoire sur l'esprit de bois et sur les divers composés éthérés qui en proviennent. lu à l'Académie des Sciences les 27 Octobre et 3 novembre 1834, Paris, 1834
Jean-Baptiste Dumas et Eugène Péligot, Mémoire sur l’Esprit de Bois et sur les divers Composés Ethérés qui en proviennent, Annales de chimie et de physique, 58 (1835) p. 5-74
Justus Liebig (1834) &quot;Ueber die Constitution des Aethers und seiner Verbindungen&quot; (On the composition of ethers and their compounds), Annalen der Pharmacie, 9 : 1–39
Jacob Berzelius, Årsberättelsen om framsteg i fysik och kemi [Annual report on progress in physics and chemistry] (Stockholm, Sweden: P.A. Norstedt &amp; Söner, 1835), p. 376
A. W. Hofmann 1866, in Monatsbericht der Königl. Preuss. Akad. der Wissenschaften zu Berlin, 1865 653