This hope comes from the fact that we don't fully understand how life originated on earth but we do know that it pretty much happened as soon as conditions allowed (geologically speaking).

So although it's a big unknown, it seems to suggest the barrier to first life might be on the lower side. So it's worth exploring anywhere that's had water (prerequisite for lots of interesting chemistry) and tectonic activity (zones of varied temperature).

Having a certain mass is also an indicator as this increases the likelihood of comet and asteroid collisions in the past, which as far as earth goes it's thought these might have carried useful molecules for life that had generated as a result of cosmic rays acting on more basic compounds on the comets surface. (NASA has found some amino acids and other pre-organic compounds on comets)

Also worth pointing out that the expectation is usually that if there are signs of life it'll be long dead. It's possible there's a window in the geological evolution of some planet/moons that makes basic replicating life not just possible but likely. That would at least offer an explanation of why earth's life arose so early. Places like Mars and Europa may have briefly supported life in the past before conditions became unsustainable. It's worth checking just in case it's still going!

We don't look for life elsewhere yet. There are some secondary efforts to say what might potentially be life elsewhere. There is no big push to find or search for life in any of the recent or past missions from NASA or anyone else.

Here is the thing, talking about finding life is easy and relatable for more simple people that cannot comprehend more scientific objectives and information. Information about finding life gets over communicated as a result.

Take for instance the Mars sample return catastrophe. If searching for life were a thing that was being done in situ returning samples would be less of a thing. People come up with lots of clever ways to say perhaps life... but not because it was the goal.

The same applies to all interstellar world catalogs. No one is searching for Earth 2.0. The Kepler mission was designed to barely resolve an Earth like analog if you were to squint, but the launch went terrible and Kepler barely made it. The launch was rough and damaged the satellite. It didn't resolve anything close to Earth size worlds around G-Type stars. There were a few random noise candidates chosen to criminally call the mission a success. If you look at the actual plot of discoveries, these points of random noise are massive outliers. Kepler was intended as just a proof of concept. It originally continuously looked at a tiny little space in the sky; one little window. Kepler only lasted barely 4 years and died prematurely. The transit method requires 3 occultations to confirm any potential world. Even if Kepler had been able to resolve an Earth analog, it would have failed to confirm a Mars orbit in just 4 years and an Earth would have had to be perfectly aligned with the respective system's orbital plane. The extended Kepler 2 mission is a joke of short orbital period objects like red dwarf systems and super large worlds. No other survey mission has even been funded to search around G-Type stars for Earth like worlds. So all the junk news you hear about distribution of planets and the oddity of the solar system are all nonsense. There has never been a reasonable attempt made to survey the sky for an Earth analog. Without the map to say when and where to look for such a planet, more powerful telescopes are useless. Something like JWST is totally irrelevant in such an effort without a map and timetable. When Kepler failed, Science failed to call out the failure and caved to political stupidity. There should have been a fleet of Kepler like surveys done thereafter, but no one has made any effort whatsoever. In this area, there was massive opportunity that was wholly squandered.

This is the truth of the search for life elsewhere. It is virtually non existent and always has been. Humanity is more than capable of creating the surveying hardware. We may have a sample size of one, but it confirms exactly what kind of star and world can support life. We can learn about life from within the Sol system, and my intention is to illustrate by talking about Kepler. We absolutely could search for life on many levels, but we don't. We do other mundane science, and individuals try to find ways of asking and answering questions about life as more of a peripheral endeavor.

The question becomes what kinds of mixes and environments are conducive to life. We don't know. So what kinds of ingredients are absolutely required for all life to exist in the presence. This is the basis of saying water is required. All life as we know it requires water and energy potential to exploit. The finer details are largely unknown to various extents.

The main reason for searching for water is not for life. It is for an exploitable future resource in missions. Water can be cracked with electrolysis to make rocket fuel.

I'm not a scientist, but I think the question you're asking is the question they're seeking to answer by searching for life in these places. There isn't an assumption that if it had liquid water it may have had life (that statement alone inherently isn't an assumption), they are searching for any evidence of life on Mars because A) it's the closest and easiest planet to get to and B) while we know in what conditions life on Earth arose, we can't be sure the same conditions are necessary somewhere else, for life that possibly may not be carbon based.