It is often more efficient to run multiple small companies under combined bookkeeping; that’s what things like classes are for in QuickBooks. I’ve always found it best to have one generic company name on bank accounts, insurance policies, etc. and then file whatever DBA papers are needed to let me, say, put checks to “Goofy’s Kidz Books” in the “Serious Publisher Inc.” bank account.
Especially with something like QB properly set up and carefully used, there’s no need to have three or five or even two little separate physical accounts for each kind of service.
If you’re really casual about things, don’t keep tight records and do things like cash checks made to your business, you’d best keep it all very separate instead.
Checks used to be a self-standing financial unit with needs and standards not far removed from Federal Reserve notes. It’s been a decade or more since they were anything but brief notes of amount and payee, with all the real validation and funds transfer happening elsewhere.
You will have chosen a name for your company when you registered the company. There’ll be a process for changing that name to another name, involving lodging more paperwork with the corporate registration people in whatever jurisdiction your company is incorporated it. If you do change the name of your company, then you need to tell the bank (and probably give them a copy of whatever certificate you get from the state authorities confirming the new corporate name).
Of course, you may not be changing your corporate name, but simply adopting a trading name under which the company will carry on business. So it would be ABC Inc, carrying on business under the name XYZ. Almost certainly, in your jurisdiction, if a person or a company is carrying on business under a name which is not their actual, legal name then there’s be a requirement to register the business name with the state authorities (so that somebody doing business with XYZ can easily find out, if they care to, that they are actually dealing with ABC Inc.) Again, if you adopt a new business name, you need to let the bank know. Again, the bank may want or may like to have a photocopy of your certificate of registration of the business name.
In most jurisdictions there is no objection to the same individual or company having two or more business names. So UDS Publications Inc can publish the Daily Gossip, the Weekly Advertiser, the Monthly Theological Review and the Quarterly Pornographer, and do business under (and receive cheques made out to) all those names. There is no reason why the bank would not be happy for UDS Publications Inc. to operate a single account, and lodge all the cheques to that account, but they probably will like to have copies of the paperwork showing that they are all registered business names of the accountholder. And they probably will be happy to issue four chequebooks, all bearing the same account number, but each bearing a different business name, so that someone expecting payment for a learned article submitted to the Monthly Theological Review does not receive a cheque bearing the name of the Quarterly Pornographer.