Drow sign language isn't an officially included language in D&D 5e rules, at least not as far as I can find.
The languages a PC knows are determined by their race and, save for a special few, are mostly listed on page 123 of the PHB. You'll notice that's there's no mention of a drow sign language. It's not even mentioned in the Sword Coast Adventurer's Guide (a 5e sourcebook for campaigns set in the Forgotten Realms).
If you also look at the references for the Wikia page you've linked you'll notice that it's mostly referencing novels by R.A Salvatore set in the Forgotten Realms and previous editions of D&D. In general, a novel does not equal a rules books for a TTRPG. Even the other question from this site that you've linked is asking about 3.5e, not 5e.
In terms of how much influence a novel does have on rules created for a TTRPG, well, that will be determined by your DM and the world they set their campaign in.
However, this doesn't mean your drow can't know drow sign language. It just means that, by default, it doesn't know it because it's not a part of the rules.
A DM is free to create their campaign setting, their world, their universe, their multiverse, however they please. If you want your drow the know sign language, talk with the DM; see if it's something they think would make sense in their world and if they can incorporate it.
The DMG even has a section on languages, from page 20-21:
When fleshing out your world, you can create new languages and dialects to reflect its unique geography and history. You can replace the default languages presented in the Player's Handbook with new ones, or split languages up into several different dialects. [...] You might invent additional secret languages, besides Druidic and thieves' cant, that allow members of certain organizations or political affiliations to communicate.
The spell does what it says it does.
The spell does not say that it translates. There is no mention of the caster having any influence over language used or that the native tongue of the target matters.
Others can simply understand what you say, regardless of the language you use to say it. They can still hear you say it, though. So if a listener does not understand the language you use, they will have the experience of understanding sentences in a language they might never have heard. I would count that as an obviously magical effect in most cases.
Best Answer
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.