This question caused me to think of another question.

The books/movies establish that Voldemort (as well as Harry) has a wand with the incredibly rare Phoenix feather as the core. Voldemort's wand is also depicted looking like a creepy bone type design (made of yew) to match his "evilness" which was made by Ollivander (at least in the movies)

Is this the same wand that was given to Riddle before his first year at Hogwarts? If not, what kind was it? It would seem odd that he would get such a wand designed like that as a child.

It's the "same" wand, just redesigned.

None of the movie wands were special-looking in the first two films. Prisoner of Azkaban introduced the personalized wands, and the trend remained throughout the subsequent films. Before that, all wands pretty much looked identical - a simple black or dark brown wooden stick. So that's why they look different - there was a redesign of nearly all the wands (out-of-universe).

As pointed out in CreationEdge's answer, the wand used by young Tom Riddle is a simple, dark-colored wand. Just as Harry's wand is the "same wand" in all eight¹ films even though it looked different in the first two films, I would say that it follows that Voldemort's bone-looking yew wand is the same wand seen being used by Tom Riddle in Chamber of Secrets. The reason it looks different is simply that it too was redesigned, just like Harry's was.

We know that Voldemort's bone-looking wand is the same one he got from Ollivander. See Au101's answer for quotes from the book.

A possible in-universe explanation could be that, at some point after Chamber of Secrets, Voldemort either had a newer, more evil looking outer shell added to his wand (or had the core put into a new wooden shell). Lucius Malfoy's silver snake head was an ornamental add on, so maybe Voldemort did something similar. :)

¹_{Yes, I know Harry's wand broke in Deathly Hallows part 1 but it was mended (off screen) in Deathly Hallows part 2, so one could still say it's the wand he used in all eight movies. Also it's simpler that way. :)}

I'm not sure whether you're asking about the books or the movies. You mentioned the books a bit in your question, but the tags seem movie-centric.

CreationEdge's answer is excellent as far as the movie goes, but in the books I really don't think there's any question that it's the same wand. When Voldemort discusses the Elder Wand with Snape in Deathly Hallows he says:

'It has not revealed the wonders it has promised. I feel no difference between this wand and the one I procured from Ollivander all those years ago.'

Harry Potter and the Deathly Hallows - p.525 - Bloomsbury - Chapter 32, The Elder Wand

When Harry interviews Ollivander in Shell Cottage, he says:

'The Dark Lord,' said Ollivander, in hushed and frightened tones, 'had always been happy with the wand I made him - yew and phoenix feather, thirteen and a half inches - until he discovered the connection of the twin cores.

Harry Potter and the Deathly Hallows - p.401 - Bloomsbury - Chapter 24, The Wandmaker

Of course, the fact that Harry's wand and Voldemort's share a core comes up throughout the series and there's never any mention of him obtaining a second wand from Ollivander, indeed we see throughout the series that the wand chooses the wizard and wand and wizard learn from each other. The Elder Wand excepting, people do not usually trade wands in.

We know that Voldemort's yew wand was the one he was using by the time he came to kill Harry and I see no reason to believe that he acquired this wand some time after starting at Hogwarts with some other wand, in fact I see every reason to doubt it.

Based on the original nature of the question, which seemed to be referring to simply the physical design of the prop within the movie:

No, he didn't have the same design. When you see his in the Chamber of Secrets flashbacks where he's confronting a young Hagrid. It appears to be a simple, sleek black or dark brown wand, with a simple handle.

There is a lot of evidence in the books that Voldemort never changed his wand once until he discovered the twin cores and the problems they were causing for him (see quotes above).
However, this very fact is a plot hole, because it is unclear how Voldemort could regain his old wand after being reinstated into a body in Book 4. There is a gap of 13 years in which we don't know what happened to this wand, although we can be sure it is the same one due to the Priori Incantatem scene (Voldemort's wand regurgitates the last spells cast from it shortly after his resurrection, and those spells include the murders of Harry's parents which obviously happened before his downfall).

There is a blog post about this interesting question right here: https://nerdpiphany.wordpress.com/2018/02/10/peeves-pets-voldemorts-wand/
To summarize that post: When Voldemort tried to kill baby Harry, the killing curse backfired, he lost his body and escaped to Albania. In Book 1, Voldemort says that during that time in hiding, "any spell that could have helped me would have required a wand", as in, he didn't have one - or a body to hold it with. So his wand must have fallen to the ground the moment Voldemort lost his body. How it got from there into Peter's possession 13 years later is an unexplained mystery. The blog post walks through several possible explanations and concludes that none of them make sense.

I had this thought that as the witch or wizard grows, their wand will change and come to reflect their personality so by the time they leave their school they will have a unique wand that lines up with them. THis theory allows for Tom to have both styles of wands without any need for convoluted explainations.