Need help with a faucet
 

Here is my bathroom faucet. I am not sure of the brand and if I could figure that out I might be able to research my problem. As you can see the handle is loose and rocks back and forth. I expected to see a small allen head screw on the faucet that I would loosen and then reseat the faucet. But there isn't. If I push down on the handle it does not seat securely and the problem remains. Any ideas would be appreciated. Thank you.

Sorry...
I looked through the current offerings for Moen, and could not find a comparable handle form.
Also the Moen have the set screw like you expected.

Are there other similar vintage faucets in the house that might reveal a manufacturer? If you don't have a regular plumber, this may be an opportunity to start a relationship so you have a guy to call when you REALLY need one.
Beyond that, I'd try to identify where this came from via previous owner or previous tradesman. If I were DIY'ing, I'd shut off the water and remove/replace the faucet (always a good excuse to buy a faucet wrench or other tool not already in your shop). Once removed, you might be able to solve the issue with a few taps from a 3 lb. Chevrolet tool. LOL.

Try turning the base piece that is below the handle. I think it might unscrew and allow you to lift it and the handle upward. You should then be able be able to see what the problem is.

Search
 

Take a good photo of just the faucet, without your hand in the way and do a google image search. It may take a couple of different views but there is a good chance you will get a positive hit

Thanks all of you for your help. I found it. I'm now about ready to crawl under the sink!

https://resources.kohler.com/webasse.../1144560-2.pdf

SUCCESS! Thanks for all of the help. Man, I would not want to be a plumber, always squeezing into these awkward positions in tight spaces. Hats off to plumbers!

solved?
 

So, what was the actual problem? Did you fix it or just replace the faucet? Did Kohler replace it under a "Lifetime Warranty"? I have a Delta kitchen faucet that clogs every several years due to mineral content in the well water. Delta sends a new one in the mail whenever needed. (2-3 times in 40+ years).

You were on the nice side of the plumbing.

The waste side, especially in basements and crawl spaces where the critters are plentiful, is even more fun!

Sorry, looking at the picture, I didn't pay attention to the fact you were using a a widespread. I should have thought of the retaining nut under the counter top as a way to tighten the entire unit up.

Many/most do not know of this trick. Nor how to do it.

I was able to release the fixture from below then take it apart. The screw that held the handle was under the faucet and just needed to be tightened.

All's well that ends well!