I guess yes, there is difference in amino acid sequences of $V_L$ and $V_H$. And so we have 6 different complementarity determining regions (CDRs) per monomeric immunoglobulin as two heavy chains are identical, and two light chains are the same. Is that correct?
You are correct. In an IgG antibody, there are a total of four protein chains — two identical heavy chains, and two identical light chains, each heavy pairing with a light to form one of the two "arms" of the molecule. Each heavy chain has three CDR regions — H1, H2, and H3 — as do each light chain (L1, L2, and L3). When the full molecule is assembled, three "full" CDRs are formed — 1, 2, and 3 — from the "partial" CDRs in the respective heavy and light chains.
You can see on the left the six individual heavy and light CDR loops separately, and on the right they are combined into the full CDR1, CDR2, and CDR3.
Here is a slightly more molecular view: