Not quite
Nazi Germany was a very odd type of state: on the one hand, it espoused socialist (in the
real meaning of that term, not the modern American one) values, but on the other, it was strongly capitalist in giving the large industrialists and major commercial companies pretty much anything they wanted in return for their support. (That last bit was mostly pragmatism on Hitler's part, and one of the reasons why people like Röhm had to go once the NSDAP was firmly in power.)
American vehicle design was largely centralised, with the US government awarding contracts to companies to make the same basic design but often tailored to their specific abilities. Which is why, say, 2.5-ton truck all look similar but differ in their details, or why a dozen companies all produced Shermans tanks.
German tank design was initially centralised as well and worked much the same way, until they began doing competitive design for things like the Tiger and Panther, and
then had the winning design built by several companies much as before. Where German procurement really differed from the American was in softskins: once the war was underway, they pretty much bought just about anything available in a slightly militarised form (much like the UK, in fact). Which greatly hampered logistics, and even such things as pushing stuck vehicles out of the mud … (because of unexpected issues like bumper heights varying widely between manufacturers).