No you should not assume he knows the language if you do not get it from your race or background. If there is no rule giving you the language you do not have it.
Keep in mind that your character need not come from the Underdark so he might in fact not know the language. (Surface Drow are a thing in some settings.) He might also come from the Underdark and never bothered to learn the language of the "lower races" living there. (In some settings Drow are quite racist.)
If you have a valid reason to know the language you might bargain with your DM to get it. You might give up one of your languages but this would be hard as you only have Common and Elvish. Giving up Common would be bad as you would probably need it to talk to the other group member and giving up Elvish would also be strange. This would basically mean creating your own custom background. This is legit but needs the OK of the DM and should not result in advantages for you.
So you should take a look what you could deal in to get Undercommon if you really want to have it.
If your DM has a nice day and you have a valid reason to know Undercommon he might give it to you for free of course. But having something of to bargain would help.
I would only hand out the language for free if I do not plan to go to the Underdark any time soon with my campaign. That would reduce the language to a style thing witch would be for free.
XGtE is optional
It's important to remember that most of XGtE's content comprises optional rules, and this particular section on character generation is no exception. In fact, the opening on page 61 explicitly states:
IDEAS, NOT RULES
Even though these pages are full of tables and die rolls, they don’t make up a rules system — in fact, the opposite is true. You can use as much or as little of this material as you desire, and you can make decisions in any order you want.
Stretching that a little, we could infer it to mean that despite what might be implied by the options presented here, none of it should be taken as informing the actual rules of the game. It definitely doesn't help that this section of Xanathar's does contradict the racial history given for tieflings in their own description in the PHB. But if we want to take the content as presented and try and make it work, we could consider the variability of genetic expression.
All cambions are half-fiends (but not all half-fiends are cambions)
Simply put, though a cambion is a half-fiend and is always the result of union between a fiend and a humanoid, the union of a fiend and a humanoid does not always result in a cambion. Sometimes, when a devil and a humanoid procreate, perhaps the result looks more like a tiefling.
To draw a crude analogy to the real world, children of mixed race parents can vary wildly between strongly expressing the racial characteristics of one or the other parent or appearing somewhere in between - depending on exactly what random bits of genetics get passed on and in which combination. It's not even unusual for children from the same parents to appear to be completely different races!
Obviously the genetics of a fantasy world are much more complicated than ours could conceivably be (especially when you allow for magical factors). It is not much of a stretch to imagine that the children of a devil/humanoid pairing may sometimes express much more of their fiendish heritage (producing a cambion) and sometimes less (producing a tiefling).
What's the difference?
The general difference between a tiefling and a cambion is that a cambion is a very strong expression of fiendish ancestry that perhaps you can only get with one directly fiendish parent, but a tiefling is a much lesser expression of such ancestry which can also occur in descendants far removed from the original fiendish influence.
Best Answer
P.258 of the DMG describes Drow Poison as one of their poison examples. So "why don't Drow in 5e, from the lowest pleb to the highest priestess, use poison?"
They do, as mentioned in the Monster Manual on p.127, and in Out of the Abyss:
(possible PC spoilers)