Given 2 conic section of type
$$ax^2+2hxy+by^2+2gx+2fy+c=0$$ have 4 intersection points. Are the points for sure concyclic?
If yes, does the proof come from the fact that I can create an infite amount of new conic section using the linear combination of them and there is for sure a value of lambda that will give me a circle?
No. In geogebra varying coefficients you can easily find counterexamples.
This has solutions
$(1,0),(\frac1{\sqrt{3}},\frac1{\sqrt{3}}),(-\frac1{\sqrt{3}},-\frac1{\sqrt{3}}),(\frac37,-\frac87).$
Which you can check are not concyclic: $(x-1/6)^2+(y+1/6)^2=13/18$ does not contain $(\frac37,-\frac87).$