Although it is very early to judge about Götze's career, I would not consider both his and Podolski's career trajectories overly similar. They are also very different personalities, which seems quite important to me:
Podolski came up through FC Cologne's youth system to become not only their go-to guy, but also some kind of über-popular folk hero for this very self-centered city. Along with his club success, Podolski quickly established himself as a fixture of a then rebuilding German NT, being perhaps the most noticable among a group of upcoming youngsters (ahead of Schweinsteiger, Lahm and Mertesacker). When Cologne got relegated, he moved to Munich, which many deemed not only a challenge with regard to sports, since noone would describe Podolski as overly cosmopolitan.
He had a disappointing stint in Munich, however I consider his next move to be his worst career decision: Podolski went back to Cologne, where he was viewed as some sort of saviour of the club and his salaries had to be financed by third-party grants. I think a move to another German club with less distractions would have been a better decision for him, but we will never know. Anyway, he had at least one very good season back in Cologne, but in the end the club could not advance from mediocrity and was even relegated again.
With Arsenal, Podolski played better than I expected at first, but his impact waned. His stint in Milan was horrific, but he seems to fit in well with Galatasaray now. But keep in mind, we are talking about a player who already many considered to be a mere mascot on the 2014 WC squad, aged only 29 then. Increasing his social media activities while becoming less relevant as a player provokes more spiteful comments than praise now.
On the other hand, Götze is a less folksy person. He excelled at Dortmund, but more as one among an ensemble of very good players than as a go-to-guy. In the German NT he never became a key player (which was extremely difficult at that time, of course) and still seems as some sort of surplus talent to me. Unlike Podolski, Götze's struggles with Munich could also be attributed to Guardiola to some extent, although he was a regular starter last season. While Götze is no impact player at all, his appearances were mostly quite strong (albeit against lesser opponents lately) and the disappointment over his Munich stint is based much more on failed expectations than weak performances than it was with Podolski.
Although I don't believe that he will be shunned completely, Götze faces a very difficult year in Munich. But after that, he will be able to move to any club he wants at an age of 25. I am sure that this will be a better decision - financially for sure and probably careerwise as well - than joning mediocre and overambitious Liverpool now.
Gruß,
BSHH