If I understand correctly, the concept of the LUCA (last universal common ancestor) is based on the hypothesis that archaea and bacteria share common ancestry.
In the realm of mathematics, the same discoveries have often been made more than once, in different places. Sometimes (like with calculus) these independent discoveries were made almost simultaneously. Sometimes the discoveries appear to be totally unrelated (Hipparchus affirmative compound propositions before 100 BC, and David Hough's work on inserting parentheses in 1944).
If "life" started on Earth in some kind of primordial soup, it is conceivable that it started not once but many times, in different places and at different times.
How strong is the evidence that bacteria and archaea do not represent quite separate discoveries of "life", which (because of the nature of the chemical substances that they use) happen to share a number of features, such as DNA?