If Galileo had no idea of Newton's Second Law or his Law of Gravitation, how did he figure out that objects of dissimilar mass fall at the same rate of acceleration if their initial conditions are the same? Only by experiment?
Galileo argued that earth gravity causes uniform acceleration. That is, Galileo argued: as an object is released to falling its velocity starts at zero, and then increases proportional with time. In other words: in double the time you get double the velocity.
My knowledge of the history of this is not sufficient, but by the looks of it Galileo did not press to generalize to the effect of any force.
In retrospect: it wasn't even necessary for galileo to assume that gravity is a force. To argue what Galileo wanted to argue it was not necessary to address the general concept of force.
Galileo argued the point that he needed to argue: that earth gravity causes uniform acceleration.
As to objects of dissimilar mass falling at the same rate. For that Galileo argued in the form of a thought demonstration, a thought demonstration that draws on everyday experience.
You can tie two bricks together with a string. Will that package accelerate in the same way as the individual bricks? If you believe the package will accelerate faster, then try it again, this time with more slack in the string. With thought demonstrations like that, appealing to everyday experience, Galileo argued that independent of the bulk of an object gravity always causes the same acceleration.