I searched on Google for both "chaill é a shaol" and "chaill sí a saol" and got little to show for it.
Searching Pota Focla for "lose life" gave three results, all using "beatha" to mean "life"; foclóir,ie has this one example under "life":
"he lost his life - fuair sé bás, bhásaigh sé, cailleadh é, d'éag sé"
with "saol" not appearing in any off the given alternatives.
"saol" seems to refer more to the period of life (lifetime), way/condition of life (lifestyle), achievements or experiences, whereas "beatha" is the state of being alive or "the fact of being alive", as Pota Focal notes under "beatha".
But even using "beatha",and variants, with various forms of "caill" didn't turn up many results.
Irish has very colourful expressions for dying. To lose one's life. To meet death. We do say these in English too, the first one anyway, although I think of that being used more in reference an accidental death or a violent death through war or whatever, rather than a "natural" one. Does this have the same sort of intent in Irish, or is it a common way to say that someone has died?
"her grandmother died" - fuair a seanmháthair bás
"her grandmother lost her life" - chaill a seanmháthair a saol
The two different idioms exist for a reason - there are circumstances in which one is more appropriate than the other. If you want to impoverish your language by only learning one of the idioms, go ahead, but Duolingo wouldn't be doing you, or any other learners, any favours by encouraging you.
The speaker pronounces “saol” as “seal,” but the spelling reminds me of the English word “soul.” Since we’re talking about “life,” would “saol” be somehow related to “soul?” I can’t see how it would be even remotely related to “seal”....unless in some folklore seals represent souls....
Sharing two consonants doesn't make two words in different languages "false friends". "soul" and saol, as you point out yourself, don't sound similar. You don't normally see the word saol and assume that it probably means "soul", whereas when an English speaker encounters "embarazada" in Spanish, and assumes that it means "embarrassed", or encounters "gift" in German, and is surprised to find that it doesn't mean "gift" in English, because those words are "false friends".
I'll admit that in this unusual example, where Chaill sí a saol appears to be a literal translation from English, "soul" isn't an outlandish connection to make, but you can see from the FGB entry for saol that this example is misleading, at best. If you were to use caill in this sense, beatha is the more appropriate noun to use, rather than saol.