Both of your sentences are grammatically correct, but they have slightly different nuances in meaning: "My husband has the same type of car as mine." This sentence means that your husband and you have identical or very similar cars in terms of type or model. It emphasizes the similarity between the two cars. "My husband has a car of the same type as mine." This sentence also means that your husband and you have cars of the same type or model, but it doesn't necessarily imply that they are identical. They could be similar but not necessarily the same make, model, or year. It's a bit more general in its description. So, the choice between the two sentences depends on whether you want to emphasize the exact similarity (first sentence) or just the general similarity in type (second sentence). Both are correct; it's just a matter of the level of detail you want to convey.

Here you can find the answer of a native speaker to a similar question on another site: (...) Related Which is correct, "She has the same car as me" or "She has the same car as I"? Which is correct, "She has the same car as me" or "She has the same car as I"? This is one of those situations in which it’s best to reword the sentence because, of the choices you’ve given, the correct one sounds awkward to many people. Colloquially, most people say “…as me,” just as we’d answer “Who is it?” with “It’s me.” That’s technically wrong, though: it should be “the same as I” and “It’s I.” But both of those sound stuffy, as if the speaker is trying to show you how smart they are. So, what to do? Find a tertium quid:¹ a third option! ▸ She has the same car as I do. ▸ Her car is the same as mine. ▸ She and I have the same car.

In my opinion those sentences are not correct. My husband has the same type of car as me/as I. My husband has a car of the same type as me/as I. "mine" in your sentences is a possessive pronoun, while "me" and "I" are personal pronouns. But: My husband's car is of the same type as mine.