Is there any address monitoring application with support for multiple cryptocurrencies?
The best option for monitoring your accounts without logging into your wallets is to use block explorers. Blockchain data is public and block explorers are made to explore this public data: blocks, transactions, accounts, account balances. 
Here are a few:

Bitcoin, Litecoin, Dogecoin:
https://live.blockcypher.com/

Ether, ERC20 tokens:
https://etherscan.io/

WAVES, Waves assets:
https://wavesexplorer.com/

ZCash:
https://explorer.zcha.in/

Most of these block explorers also have API solutions.
I have created a playlist showing how to use the zChain ZCash block explorer API here: Zchain ZCash block explorer API - INTRO