Downloading TubeMate is legal, not prohibited. What you do with the downloaded videos is another matter, as they may infringe on intellectual property rights. In the end, the responsibility for the downloaded videos lies with the end-user. The difference between watching a streamed or downloaded video is minimal. From a section of the application, the developers warn that the downloaded videos are intended for personal use and enjoyment, but not for distribution or commercial use.
On the other hand, there is the YouTube warning that says you cannot consume videos outside its platform, that is, it would not be possible to download videos. That is why you cannot find it in Google Play. We must resort to alternative stores where you can find TubeMate YouTube Downloader. Websites like Malavida offer to download the APK with total security for your mobile device. The same happens with other applications that, due to Google's restrictions, are not in the official store such, as Vidmate, Snaptube, and other variants of official applications.
It should be noted that the developers or programmers, or any professional in the sector, live from this type of apps that they publish and maintain so it is very important the proper use to maintain this application.
As always, the end-user is the one who decides what to do with the videos or audios that are downloaded. If it is for private use there is no problem, but if we commercialize or distribute these videos, these are big words and we would be violating the copyright.