Actually its part of the lute family...
It all depends, as usual, on the definitions you have for your categorizations. We all have such definitions, but in most cases we use them unconsciously.
Anyway, ethnomusicologists have developed a system of instrument classification based on the physical principal used to produce a sound, i.a. does a string vibrate, some air inside the instrument or a skin, or even the whole instrument?
To sound more scientific, they use the Greek words, such as chordophone (for stringed instruments), aerophone (for wind instruments), membranophone (for instruments with skin or membrane) or idiophone („self-sounding devices“ such as rattles or shakers or scratched things like a güiro, or xylophones).
Of course, you can create percussion effects on a uke, especially on a amplified one, but I think, we all can agree on the fact, that the typical way to make a sound with a uke is by plucking the string, so it‘s a chordophone.
After that, traits of the construction are used to diferentiate, like what does the body look like, or is there one at all, in the first place? East-Asian string instruments are built with a simple, but curved board, without any proper body and neck. Harps are shaped more frame-like, also without a neck.
Of course, many instruments developed during the 20th century somehow defy theses categories, like the electric guitar or the synthesizer. They are often put into the „catch-all the things we cannot properly categorize“-category of electrophone.
As far a I know (but I did my MA in ethnomusicology some 15, 16 years ago, so I might not be up-to-date here...), the most commonly used classifications in museums and instrument collections was developed by German early ethnomusicologists Curt Sachs and Erich Moritz von Hornbostel. Both them did the bulk of their scientific work pre-WWII and both fled from the Nazis to the USA in the 1930s. Of course, there are other systems of classifying musical instruments, but this one was one of the first trying to be scientific:
https://en.m.wikipedia.org/wiki/Musical_instrument_classification
According to Hornbostel and Sachs, stringed instruments with a body (made of wood or a gourd or any other object) and an attached neck is called a „lute“. THus, I would sum up this whole discussion: John Colter is right, LarryS, too, and so is myself, a little bit: the uke and the guitar do indeed belong to the same family/category of instruments. You may call it anyway you like, this category, but often it helps to use well-established and/or widespread categories, if you want to make sure the people you are talking or writing to get your point.
So, have fun, y‘all, with plucking your little guitars/lutes/ukes/chordophones/tiny stringed instruments!
EDIT: Out of curiosity (sometimes, I just can‘t help it...), I checked again on this site:
http://https://en.m.wikipedia.org/wiki/Hornbostel–Sachs
According to Hornbostel and Sachs, the guitar, as well as the uke (but also the violin, for example), the category for our beloved plucked little string instrument is „necked box lute“. Of course, it has a number, scientific classifications have to have numbers, in order to look really scientific. The number of the uke is 321.322. I have no idea, how this might help, though...