No, they're really not. True tankless on 1500W (max plug-in power) is actually a hard problem, because physics is against you. You cannot deliver enough BTUs to do the job.
That's why they're not much of a product.
The common ones are not tankless.
Most "under sink" heaters sold are not tankless at all, they have a little tank about the size of a 2-liter Coke bottle which they keep pre-heated. Those are small, which is a requirement you have, but run much too hot for your use. They are for tea. Or they have a 2.5 to 6 gallon tank for washing, but those are too large for your application. I have a 4-gallon at the lodge; it's over a foot in diameter and 18" tall.
Note that the compact "tea sized" tankless impede flow too much to be inline, and their minimum setting is far too hot. The larger 3-6 gallon tanked heaters have acceptable flow for main-tap use, but are quite larger than you have room for.
Can tankless work on existing wires?
You have the constriction "I don't have a 220v and don't want to spend the money for an electrician to put one under the sink". So what if we don't? What if we try to work with the wires in the walls? TLDR: impossible at main-tap flow rates, but you could use a "side tap" to limit the water flow to that which would be possible.
You can do the math. It takes 1 BTU of energy to heat 1 pound (pint) of water 1 degree F. So if you want to warm 1 pint from 50F to 90F, that is 40 BTU. Now think about your flow rate, how many gallons per minute do you want to flow? A gallon is 8 pints and 8.3 pounds. The faster the flow, the more BTUs you need.
1 watt for 1 hour gives 3.41 BTU.
1 watt for 1 second gives 0.00095 BTU.
Most of those heaters are 1500 watts. 1500 watts can give 1.4 BTUs per second, or 85 BTU per minute.
Now you see plain why this isn't going to work on main-tap flow. At 2 GPM flow (16 pints/min) you are dividing 85 BTU by 16, and getting about 5 degrees of heat rise.
However, as a side-tap it could be a player - at 0.25 GPM flow (2 pints/min) we are 85/2 = 42.5 degree rise. At 0.375 GPM (3 pints/min) we have 27.5 degree rise. Both are sad even for a side tap, but better than nothing.
Now if the heater has a dedicated circuit just for it, then it cannot be overstated how easy it is to convert a 120V circuit to a 240V circuit (at the same amperage!!!) 15A wire (#14) can give 3600W, 20A wire (#12) can give 4800W. Crunch the numbers based on your inlet/outlet temp and desired flow, but it seems like a respectable side-tap could happen - 3600W giving 48 degree rise at 0.5 GPM (still side-tap flow).
We're still quite some distance from it being viable on main tap.
What if power were no object?
If you want tankless to work on the main tap at normal flows, power cannot be an object. But you must balance the heat rise you are willing to accept with the energy required to get that rise.
Further, the amount of power will be a huge fraction of total electrical load in the house (really). As such, it will require a standard NEC Article 220/CEC section 8 Load Calculation to determine whether the existing service can handle this new load. If not, then either a service upgrade will be required, or new "demand-side management" tech that has not quite arrived.
The designer carelessly undersizing a tankless system is the #1 way in which tankless heater installations fail. Invariably, the half-wit DIY-designing homeowner blames tankless tech, when the real problem is in the bathroom mirror. I hope I inspire an appropriate disdain for this careless character.
So let's crunch the numbers. My own sink flows 1.25 GPM, however it's a shadow of its former self. I would aim for 2 GPM out of an abundance of caution. Let's figure the wattage per degree rise on 2 GPM. Well, 2 GPM is 16.6 pounds per minute, so we need 16.6 BTUs (absolute) per degree rise per minute. That is 16.6*60 = 996 BTUs/hour per degree rise. Divide by 3.41 BTU/watt-hour and we get 292 watts per degree of rise at presumed worst-case 2 GPM for a kitchen sink (not a whole house).
So at 2 GPM if we want 10 degrees rise, 2920 watts. 50 degree rise, 14604 watts, and at OP's guess of 55 degree rise, 16000 watts given assumed 2 GPM flow since OP won't say.
(and those >6kW tankless start to get large, so back to the space problem).
Yeah, that's a lot. This is the tyranny of tankless water heating. And like I say, this is where homeowners cut corners and put in too small a system.
So how in blazes do the British do their "electric showers" on a measly 40 amps (9.5 KW)? By sharply limiting flow to 1 GPM (141 watts per BTU) or less. Low-flow even by California standards. But that's not going to work on a general tap unless constrict the tap flow all the time.
Unfortunately in comments you have precluded every one of them, so I don't know what to tell you lol.
- Use a tanked heater inline. However, the small/compact heaters (2 quart) are too low flow, leaving the 3-6 gallon tanks. Those take a lot of space, and as you say "I don't have enough space to install one because I have a water filtering system with a tank which used a lot of space in the cabinet." So cross that out.
- Use a side-tap. This would allow you to limit flow to a rate low enough (0.25 GPM) for a small heater like an EEmax Spex1812T to work. However you have said "The water is going into the same main faucet where it gets mixed with cold water to produce warm water." so cross that off.
- Bring big power. However the amount of power is considerable, and this requires a Load Calc plus service upgrade or demand-side management. Also, you say " I don't have a 220v and don't want to spend the money for an electrician to put one under the sink." so cross that off too.
What does that leave? Water circulation systems. That requires either a return pipe to be plumbed, which involves tearing up the house. Or a hinky, dangerous hack of bootlegging the cold water line to shove the cool/tepid water back to the heater. If anythign goes wrong with that, both lines will be hot, and a thermostatic mixing valve cannot protect you from scalding.
Other than that, I don't know what to tell you. If I were you, which I'm not, I would find the space for a 4 gallon tanked. I have one at the lodge and has sufficient flow and capacity to be a "reliever" until the main hot water arrives.