Yes, it's possible. Some think that Mars had an oxygen rich atmosphere early on. While life could have been responsible, it could also just be the result of solar storms. Without a magnetic field it wasn't as protected from solar storms as Earth, so the storms could split the oxygen from the hydrogen in the upper atmosphere. The hydrogen being lighter escapes. The oxygen then builds up in the atmosphere.
Indeed this process may be the reason that Mars is red, why it has iron oxides and why its salts are highly oxygenated sulfates, perhchlorates and chlorates instead of the sulfides and chlorides we have on Earth.
See Early Mars atmosphere 'oxygen-rich' before Earth's - BBC News
On Early Earth the iron was unoxidized and only went rusty when photosynthetic llife started to raise the concentrations of oxygen in the atmosphere. And for a long period of time, millions of years, the oxygen levels didn't rise far because all the oxygen was used up making the iron deposits rusty.
The oxygen caused the dissolved iron and other metals to precipitate out of the ocean as oxides creating the Banded iron formation
Banded iron formation Dales Gorge
On Mars this same sort of process may well be what turned the planet red. But need not be due to life. It could just be due to solar storms splitting oxygen out of water in the upper atmosphere and the hydrogen escaping.
So detection of oxygen alone is not enough reason to say an exoplanet has life on it.
James Lovelock suggested that if you detect oxygen and methane say, two different gases that will react together over a short timescale, in the same atmosphere, it could be evidence of life.
Even that is not foolproof. The methane plumes on Mars could be caused by inorganic serpentisation. If it was a strong signal it could be a decent indication that there might be life there, to be followed up further.
We could however also detect life directly on an exoplanet from biochemical signatures, e.g. the "Red edge" of chlorophyll. Something that suggests a prevalent molecule on the surface that seems unlikely by only chemical processes without some replication going on.
And just to say, it can of course go the other way, a planet could have no signs of life detectable from a distance yet have abundant life. E.g. ice covered planet or moon like Europa. Or an ocean world, with life on the sea bed, perhaps supported by hydrothermal vents etc, but atmosphere in equilibrium like early Earth before photosynthesis.
Or a terrestrial planet but it is dry on the surface, no oceans, but life deep underground, in an underground "hydrosphere" layer of water that covers the entire planet, but beneath rock so invisible from above. Mars may have such a hydrosphere deep underground for instance.
Or it could be photosynthetic life is abundant there, but all the oxygen is being used to create metal oxides like our banded iron formations and so far there is no noticeable impact on the atmosphere.
Indeed also modern Mars might have life on the surface, but it could have no detectable effect on the atmosphere if it is sparse, slow growing, and only occurs in special places on Mars. There's a tiny amount of oxygen in the Mars atmosphere anyway. So a signal from photosynthetic life, sparse quantities, say cyanobacteria or lichens, and other microbes etc feeding on them or living with them in microhabitats - that would not be noticeable at all. You could only find it by looking for it close up on the surface. Not even from orbit, never mind looking at Mars from the distance, from another star.
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