There's a lot to unpack in your post!
Poincare proclaims that the mathematical continuum originates from the
sensible intuition and that intuition by pure number or logic alone
could not have given us this notion.
Intuition by pure number meaning induction. Indefinite iteration.
Logic is of course the natural language of the "analyst".
In light of this, I immediately thought of algebraic geometry and
whether or not the continuum is really something that can only come
from the geometric/intuitive (excluding induction) side of things.
I think the question is an excellent question is rooted in the nature of the exploration of the primacy of the senses, and ultimately has to be approached with an empirical bent. That being said, human beings are far more predisposed to using their visual systems to their logical ones. It's a fact that even statisticians struggle to use their statistical knowledge well. (See Thinking, Fast and Slow.)
The historical fact is that the developments of continuity from the direction of intuition to rigorous analysis and moving from Euclidian axioms to non-Euclidean axioms both show a bias of the brain as ideas are generally introduced with intuition and then later formalized with logic. In the former case, the idea of continuity started essentially as a visual-kinesthetic intuition that the smoothly curved line was special in a way that other lines were not. Ultimately, analytic continuity was introduced once closure of the reals was put forth. (See Ch. 4, Numbering the Continuum of The Philosophy of Set Theory). As far as why for so long did mathematicians accept uncritically alternatives to Euclid's parallel postulate? Again, I would suggest it's a bias in how our visual system dominates in our cognition and language. There's something inherently off presuming lines that don't look parallel can be considered parallel lines.
I can hardly see how Poincare can claim the continuum and geometry have such a close relationship. Didn't people just will into existence the square root of negative one???
As for the relationship between the continuum and geometry, they are inextricably intertwined. The correspondence between points on a line and the reals is a necessity not only in understanding analysis, but also sets, and this is because what is happening in the brain when moving from the geometrical to verbal domain is a cross-domain mapping that requires development. Lakoff and Nuñez wrote a book offering a theory on what happens based on an idea called the conceptual metaphor, which is related to a scientific study of meaning itself, cognitive semantics, a field in cognitive science.
How does Poincare go from numbers, which are abstract concepts, to the
source of essentially the concept of the infinite in math being from
To answer this, try an exercise. Draw an isoceles triangle with a height greater than a base, and then draw a segment from the two midpoints of the legs. Note the segment from the midpoints will be shorter than the base. Now, put some points on the base of the traingle and draw a segment through the midpoint segments to the vertex between the legs. Then another. Then another. In fact, there are an infinte number of points and segments that can be drawn showing there is a 1-1 correspondence between the points of the midpoint segment and base. Yes, despite having different lengths, they have the same number of points. And yet wouldn't the two differing lengths represent the same continuum? Hence, questions of infinity and the reals arise from a simple Euclidian construction, in this case introducing an analytic proposition which is counter-(geometrical-)intuitive
I am apt to agree with those who have commented on your question. Different people have these mutliple intelligences, and some have lots of all of them, and some struggle with most of them. A square after all can be spoken of, drawn, measured, or represented with abstract, unknown lengths. Despite the origins of geometry and arithmetic being distinct, it is easy to see given the historical developments as well as the development of children that geometry often helps lead us to analytical propositions.