Does acid-catalyzed alkoxyation have the same mechanism as acid-catalyzed hydration?

Not necessarily. In general, they have similar mechanisms. The overall steps are:

• Use the acid to protonate the water or alcohol, and the newly-generated catalyst protonates the alkene (the water or alcohol is assumed to be very abundant to promote the forward direction in the reaction)
• Allow any 1,2-hydride shifts or 1,2-alkyl shifts to occur, if applicable
• Water or alcohol that isn't protonated, if a good-enough nucleophile, then adds onto the carbocation. If not a good-enough nucleophile, it may just grab a #beta# proton some of the time and it may act as a decently-strong nucleophile some other times. In some of those cases, it can be a mixture of products.
• If the nucleophilic attack was successful, then to some extent, water or alcohol that isn't protonated grabs a proton off of the currently-positively-charged product and reforms the catalytic, protonated water or alcohol to repeat the mechanism again.

The main difference is that the alcohol can be any type (primary, secondary, tertiary, very sterically hindered, not very sterically hindered, etc), whereas obviously water is pretty much always water (unless you deuterate it or something and it's #D_2O#, but that isn't extremely significant here).

This difference allows for the alkoxylation reaction to be not necessarily an #S_(N)2# reaction. If the alcohol is bulky (like tert-butyl alcohol), it could easily be an #E1# or #E2# reaction.