Yes, each creature will either have the statistics for the standard skeleton or zombie
When the spells say that a creature becomes a zombie or a skeleton they are referring to the respective creatures with those names in the rules. It isn't clear why they felt the need to add the part about the DM having the creature's statistics however, but there really isn't any other reasonable way to interpret the effect.
It would not be in line with any other spell effect, for example, to interpret this to meant that the DM comes up with the statistics from scratch each time for every creature that is raised. And there is no zombie or skeleton monster template, so there is no way for a DM to create individual stats for a monster-turned-undead. Really the only interpretation we are left with then is that the spell wants you to use the standard statistics for those creatures.
It is worth noting that the "the DM has the creature’s game statistics" language appears in all Conjure X spells (plus some others).1
One thing that might be different from creature to creature is equipment as discussed in this Q&A.2 Your DM might provide you with slightly different statistics based on that, but the base will always be those standard stat blocks.
Jeremy Crawford has unofficially clarified the intent on Twitter though:
Q: When animate dead is cast on a corpse/skeleton, do you use the Monster manual statistics, or add a template?
JC: When you cast animate dead, use the skeleton/zombie stat block from the Monster Manual or appendix D in the PH.
1 - thanks @Miniman
2 - thanks @Someone_Evil