Yes, local accounts are those that exist - are/have logged in - on the same console, whether created there or elsewhere.
The Playstation Plus subscription is tied to the SEN account, but that account can specify a single PS4 console as its primary PS4, which enables other (non-PS+) accounts to access everything on the console.
So any account using that console (the one specified as the primary by at least 1 PS+ subsriber) can enjoy the benefits of PS+, when using that console. That includes multiplayer.
I assume though that multiplayer is not available to local accounts if you are online and playing multiplayer on someone else's console. That part isn't exactly clear. But your (non-PS+)family members / housemates could definitely play multiplayer on your console with only your PS+ subscription when you're not using it.
This FAQ on playstation.com is fairly detailed:
PlayStation 4
You can activate one PS4™ on your Sony Entertainment Network account
(this is known as your Primary PS4) and temporarily activate a second
PS4 on your account.
PlayStation Plus benefits available to other non-subscribing users on your Primary PS4:
- Online multiplayer (subject to any parental control settings on sub-accounts)
- Play downloaded games purchased with PlayStation Plus discount
- Play downloaded IGC games
PlayStation Plus benefits not available to other users on your Primary PS4:
- Purchasing PlayStation Plus discounted products from PlayStation Store
- Online game save storage
- Auto patch download
- PlayStation Plus exclusive early access to game trials
PlayStation Plus benefits available to you on a non-subscribing friend’s PS4:
- All of your PlayStation Plus membership benefits are available to you when you use a friend’s PS4.
PlayStation Plus benefits available to your non-subscribing friend when you use their PS4:
- Play downloaded games purchased with PlayStation Plus discount
- Play downloaded IGC games
PlayStation Plus benefits not available to your non-subscribing friend when you use their PS4:
- Online multiplayer (subject to any parental control settings on sub accounts)
- Purchasing PlayStation Plus discounted products from PlayStation Store
- Online game save storage
- Auto patch download
- PlayStation Plus exclusive early access to game trials
When you sign into your Sony Entertainment Network on a friend’s PS4
the system is temporarily activated on your account. Please ensure
that you correctly sign-out of “PSN” on your friend’s system to keep
your account secure.
If your “PSN” account is inactive on your friend’s PS4, the system
will be deactivated automatically from your account and your friend
will lose access to any of your downloaded content. This content can
be deleted from their system hard drive by going to Image (Settings) >
System Storage Management > Applications. Next, highlight the game to
delete and press Options > Image (X button).
Source: http://faq.en.playstation.com/app/answers/detail/a_id/5030
Best Answer
It's part of PlayStation's wacky permissions things:
Because you are a sub-account to your mom, you have limitations to what you can purchase. PlayStation Plus is apparently one of those limitations. It might also be more than your mom let's the account spend per month. It might be that it's more money than your mom has on her account. There are a bunch of reason why a sub can't buy.
Even though you are still covered by your mom's PlayStation Plus account as a sub, she can only have one "primary" PS4 console. So while you're playing on her console, everything should work fine, but on your new console, she does not (or at least I'm assuming she does not) have an account on it. She would need to log in you your console and declare it her primary PS4. But that means the original would no longer allow her to access PS+ content.
You're only real option is to go through the process of detaching yourself from her account and paying for PS+ yourself.