I have no experience with this type of faucet, but from the picture it appears it is held in place only in the center. There are several possible ways the hinged clamp in the center may work. Is the hinged clamp in fact holding the body of the faucet or is this just a connection to the drain plug?
It may be that you pull down on the rod and press up on the clamp. It may be that you have to pinch the side tabs together to move the clamp up or it may be that the tabs are to be pressed only to remove the clamp.
It may be that you twist the clamp with or without pinching the tabs together.
The center looks almost like a toggle bolt, but with a fixed T instead of a toggle.
The drain lever is inside of a tube that appears to be the main fastener. Is that tube threaded? If it is, does it turn? If not, can the T be turned? I don't see a picture of the back side of the upper portion, but if that can be turned, it will tighten the unit. If that tube is not threaded, it may be a ratcheting connection to the T part. I doubt pressing up on the T will help much, but you might try. But if you can insert a shim between the T and the sink, that should shore up the looseness. A shim could be made from something like a popcycle stick beveled on one side, or perhaps a plastic shim used for furniture legs (Found in home depot near the furniture sliders). Trim the shim to fit the space, and wedge that in there.
Another option is to caulk the faucet with clear silicone (after making sure it is really dry). If you rub off the excess silicone after it starts to set but before it is dry, you can minimize and visible silicone. Try and force some of the silicode inder the faucet, using something really thin like a very fine feeler gauge... The goal is to take up any slack, and glue the faucet down. Also, the silicone seal will reduce water damage to the underside of the faucet, making the faucet last longer.
Had the same question but didn't find the answer here. Decided to take the drain lever out and looked in. Sure enough there was a place to turn the threaded tube with a Philips head. I loosed it all the way to clean up the moisture underneath the faucet and then tightened it right up no problem. Hope this helps someone else.