I understand that if you are a freelancer, and you care about taxes and you must issue real invoices to clients, then you may avoid the Upwork invoices and issue invoices instead with your own invoicing software, and keep those self-issued invoices.
However my problem is as a client that if I pay a freelancer, I need some form of invoice, and I cannot issue an invoice in the name of the freelancer obviously.
I think the problem here is that a lot of freelancers don't do the research and just start working on Upwork as a private person. But I guess in most countries you must register yourself at least as a sole trader, because if you are freelancing and issueing invoices, you cannot do it as a private person, you need at least sole tradership. Probably not in all countries. And if you issue invoices as a sole trader, then you must usually keep those invoices, and care about the format of those invoices, to comply with local laws.
this is what a lot of freelancers are not doing, so clients have problem receiveing a valid invoice. But this is just my theory...
As a freelancer, do you issue your invoices on your own country's system to Upwork or to the client?
As far as I know, the client receives an invoice from Upwork regarding my work. As I receive the payment from Upwork should I invoice Upwork (and not the client)?
No, as a freelancer, you must invoice the client, not Upwork. Because Upwork is just an escrow service here. And the client must receive the invoice from the freelancer.