There is no restriction on the amount of savings accounts you can get. However to get the best rate you need to constantly keep checking the interest rate on the accounts. I would say the fewer the better as this will make it easier to check to see if the interest rate you are receiving is still competitive.
As for ISAs they stand for individual Savings Accounts as the name suggests they are for individuals, I don't think you can open one in a joint name....
There are 2 types of ISAs one is for Cash one is for stocks. The restriction on ISAs is that you can only pay in a certian amount per year, the allowance for cash ISAs this year is £5,640. This is tied to your National Insurance Number. I don't belive there is anything stopping you from paying into 2 diffrent ISA accounts in the same year as long as the total between them does not exceed the limit. Although I am not sure why you would want to do this.
I would suggest you get 1 Cash ISA each first and max these out before you worry about getting additional accounts as the interest you earn on them will be tax free. So you will get the best guaranteed return from these types of accounts.
Although you can only pay in X amount of money in each tax year you are free to transfer money all ready accumulated in ISA accounts when ever you choose and this does not affect your limit for that tax year.
for example you could open a ISA account this year and pay in the maximum limit of £5640 as take advantage of one of the promotional offers avaiable (eg extra 2% for 12 months) at the end of that year you are free to open another ISA account (taking advantage of the best deals available at the time and transfer the balance from you current ISA to your new one and pay your next years allowance into that one.
Assuming next years allowance is the same as this years you could have £11,290 in an ISA account for you and the same for your partner.