To sound naturally in Polish, it could be "Ty i ja, obaj/oboje/obie lubimy kanapki." or "Ty i ja, my obaj/oboje/obie lubimy kanapki."
In other words, "my" alone does not sound well, it requires a numeral: "both": for 2 persons: "obaj" is 2x masculine, "obie" - 2x feminine, "oboje" - mixed; also "oba/obie" would be a collective numeral for non personal nouns, like 2 animals or inanimate objects (depending on their grammatical gender).
But this type of phrase with redundant subject, does not serve to give some information about any fact related to (direct or indirect) object, but to stress the similarity between two parts making the collective subject (you and me in this case).
That even sounds exactly like Russian. So, I did some research, but couldn't figure out whether this is an archaism or a Russicism/Ukrainism. I would even say that it's both. Famous Polish writer Juliusz Słowacki wrote in his Balladyna (written 1834) - "O! my z tobą będziemy szczęśliwi!"
This course has shown me that my own particular brand of Polish is a total hodgepodge mix of archaic, different regional slang, country bumpkin and uber- intellectual. Throw into that a sprinkling of totally fabricated words and phrases. That's what happens when the people you learn from are all expats (many emigrating after the second world war) and from all walks of life :-)
We often use object forms in double subjects in informal speech and I is sometimes used informally in double objects. These forms have been common in educated speech for centuries.
Here's an example of a subject form I used as an object. "I often think of the old days and how you helped Bertie and I", literally the Queen's English.