In short, no. Kakashi didn't achieve an Eternal Mangekyou Sharingan.
According to wiki:
After Kakashi loses his Sharingan, his body is temporarily inhabited by Obito's spirit, giving him Mangekyō in both eyes.
So now the question is Why he didn't achieve Eternal Mangekyou Sharingan ?
Now again according to wiki:
Their vision can be restored by receiving the transplanted Mangekyō of
an Uchiha with strong blood ties — ideally a sibling — thus
awakening so-called Eternal Mangekyō Sharingan. The Eternal Mangekyō
merge the designs of the transplanter and the transplantee's
Since Kakashi was not having his own original Mangekyo so it might be possible that he was not able to achieve it also it is mentioned that there has to be strong blood ties. So I guess according to me both of them have to be from Uchiha clan but in this case Kakashi was from Hatake clan.