went with cheap smartwatch

Well on the bright side, you didn't waste much money on it

The question is, why does your phone not allow you to install it? Bear in mind that by default installation from sources other than the Play Store is disabled, so you have to allow that before you can install and apk. If that's all it is then you shouldn't read anything into it. That said, you should only install apks from trusted sources, so you do need to decide whether you trust these people.

From what I've seen it looks like that app is designed for that company's watches, and if they are not using standard interfaces it's unlikely anyone else is going to bother making an app for them. So it probably is that app or nothing.

I'm not into smartwatches (the watch I'm wearing today is powered by a spring rather than a battery...), so not an expert. But a WearOS watch (Google's smartwatch app) will either require no app or a very standard one you can install from the Play Store. Samsung's own smartwatches will be similarly straightforward (Samsung watches usually run their own OS rather than Wear). I couldn't recommend a watch without knowing what features and design you want and what your budget is.

 

The question is, would you be interesting in buying a $300 watch, or would it be a waste of time anyone suggesting one? That's what I meant by "what is your budget?": someone might decide to try a cheap watch to see whether they got on with smartwatches rather than necessarily because the cheap watch was their only option, so I try not to make assumptions.

Whatever that watch is based on is probably less "generic" than WearOS, in that there are many WearOS watches but probably few others that use the same system and interfaces as that one. It's very possible that your watch will work (to the level you might expect for a $15 watch) if you install the software. You haven't said anything about why you were unable to do that, i.e. whether it's just that you've not enabled "unknown source" app installation on your phone.

 

That $15 cheapo, like many of these things, is anything but "generic" . As it requires some proprietary app from an unknown manufacturer, that only works with that particular watch. It also might be phoning home your private info to servers in China, for aggregation and sale to other third parties.

 

I had one of those @Pedro Vera
You'll probably get a year of use out of it.
Just use the apps on the watch. Google Play won't work with it.
They do do their primary function of keeping time very well.