My friends and I are playing Lost Mine of Phandelver and it is loaded up on the app. I was able to steal the staff of defense because we snuck into a room and I passed a sleight of hand check against someone.

https://ddb.ac/characters/67842555/X6Ay0H

I am playing a Woof Elf Monk 1/Sorc 4 with proficiencies in Longbow, Longsword, Shortsword, & Simple Weapons   My DM and I agree I should be able to both use the charges to cast shield and mage armor while being able to attack. For whatever reason I cannot attack with staff of defense unless I use the customize drop down menu in my inventory. Is the app right or are we right?

If you mean, can you use the staff like a quarterstaff, yes, you can. 
It won’t give you any bonus to hit or damage, but it does count as a staff, and therefore a monk weapon. 
Fwiw, it’s also a magic staff, for purposes of overcoming resistances. At least that’s how I’d rule it. 

Edit: Nevermind I was entirely wrong XD. See the posts from DxJxC and IamSposta below.

There are magical staffs that can explicitly be used as a Quarterstaff such as the Staff of Power. Unfortunately the Staff of Defense doesn't say it can be wielded as a Quarterstaff and that is why DnDBeyond doesn't automatically provide an attack for it when equipped.

It is common for DMs to allow staffs to be used as quarterstaffs but as always check with your DM first. Personally I agree with Xalthu but others may rule differently.

Per the DMG:

Unless a staff’s description says otherwise, a staff can be used as a quarterstaff.

 So it can be used to attack. Unfortunately, DDB categorizes staves as a different item type from quarterstaves with no weapon properties. IDK why.

Easy fix, just homebrew a copy of it and change the item type to quarterstaff.

Quote from Fangeye >>

There are magical staffs that can explicitly be used as a Quarterstaff such as the Staff of Power. Unfortunately the Staff of Defense doesn't say it can be wielded as a Quarterstaff and that is why DnDBeyond doesn't automatically provide an attack for it when equipped.

It is common for DMs to allow staffs to be used as quarterstaffs but as always check with your DM first. Personally I agree with Xalthu but others may rule differently.

 
Actually, according to RAW, almost any staff can be used as a quarterstaff:

https://www.dndbeyond.com/sources/dmg/treasure#Staffs

Staffs

A magic staff is about 5 or 6 feet long. Staffs vary widely in appearance: some are of nearly equal diameter throughout and smooth, others are gnarled and twisted, some are made of wood, and others are composed of polished metal or crystal. Depending on the material, a staff weighs between 2 and 7 pounds.

Unless a staff’s description says otherwise, a staff can be used as a quarterstaff.

Quote from ee3pnqm25f >>

My DM and I agree I should be able to both use the charges to cast shield and mage armor while being able to attack. 

 Your character should be able to use your action to attack with it as a magic quarterstaff OR  to cast one of the spells from the staff, not both since each use an action if it's also what you were wondering.

Mage Armor would take an action while Shield is a reaction. 

And yes, the OP's character should be able to use the staff as a quarterstaff to attack with or as a stave to cast the spells from. 

It's true that both spells aren't likely to be cast during your turn in combat, Mage Armor is more often after a long rest and Shield off turn.

Thanks for all the replies everyone. I get it that it's the app's fault that I cannot attack with Staff of Defense by default in the app or the web browser. My DM has allowed me to home brew it once I obtained the quarterstaff after I noticed the problem. I just wanted to know if by default the app was correct about that and since the answer is home brew we all know the app is wrong by default.

For clarification I meant on my character sheet in the app or the web browser under the "Actions" > "All" tab that I should have the option to cast Mage Armor, Shield, or have the ability to attack with the Staff of Defense without having to homebrew the answer with the "Customize" > "Display as Attack". It kinda bothers me my coolest weapon has a "Customized" label next to Melee Weapon like it's my fault I had to work around it :(

Lastly: What for whatever reason when I switch between the app and a web browser the charges don't always tic off and doing it manually has been a pain at drunken times. My iPhone died and whoops I needed to use my laptop.

Quote from ee3pnqm25f >>

Lastly: What for whatever reason when I switch between the app and a web browser the charges don't always tic off and doing it manually has been a pain at drunken times. My iPhone died and whoops I needed to use my laptop.

 
The app doesn’t update in realtime, it only updates to/from the system once at night when the OS syncs to the cloud.

Thanks! I just need to take better notes.

Another question I have on the staff specifically is does it count as a weapon for the purposes of the duelling fighting style.

I'm playing an Arcane Trickster Rogue 7 / Swords Bard 6 so my primary weapon is shadow blade.

I have the duelling fighting style but wield the staff for the AC buff and shield.

Does the staff count as a weapon since that would negate my fighting style?

Quote from Oliver_B >>

Another question I have on the staff specifically is does it count as a weapon for the purposes of the duelling fighting style.

I'm playing an Arcane Trickster Rogue 7 / Swords Bard 6 so my primary weapon is shadow blade.

I have the duelling fighting style but wield the staff for the AC buff and shield.

Does the staff count as a weapon since that would negate my fighting style?

 Magic staffs are considered weapons unless their description says otherwise.