In the left figure below the locus of the center of the circle tangent to a given circle and line is a parabola (yellow "curve"), while in the right is locus of the center of circle tangent to 2 circle is a hyperbola (green "curve"). Can ellipse be generated the same way using only circles and lines?
Yes, it is possible: just take two circles one inside the other (blue circles in diagram below), and a third circle internally tangent to the larger circle and externally tangent to the smaller circle.
Major axis of the ellipse is the sum of the radii of the blue circles, while their centers are the foci. Notice that blue circles could also intersect, provided you allow the tangent circle to swap internal/external tangency between them.