...would it be technically possible to produce everlasting (or at least) longer lasting toothbrushes?
Yes. You can use a wire brush - since it's harder than the teeth it will be used on, it will not wear out.
The teeth, however, will wear out.
The root problem is that the purpose of the toothbrush is to remove material that is softer than your teeth, without removing the hard enamel coating of your teeth. The bristles, thus, have to be softer than tooth enamel, and harder than tooth deposits.
But you end up rubbing this material up against the tooth enamel - which means it will wear down.
Anything harder and longer lasting will have a detrimental effect on the teeth you're trying to protect.
I assume that most people replace their toothbrushes every 4-6 weeks.
The American Dental Association recommends toothbrush replacement every 3-4 months, or 13-17 weeks, which is 3 times longer than your assumption.
Perhaps people can reduce their toothbrush waste by 3 times merely by changing from a 4-6 week schedule to the recommended 13-17 week schedule.
That said, I replace my bristles once a year. They are colored with a dye that changes according to use so I know when to replace them per my use, and that's about a year. Further I use a replaceable head toothbrush, which is about half the volume of materials as a regular toothbrush. These two things - infrequent replacement, and replacing only the head - may make a significant difference if adopted more widely.
This, of courses, generates a lot of plastic waste.
This depends on your definition of "a lot". A single human, even following your excessive replacement schedule, would be throwing away about a quarter of a pound of plastic per year just on toothbrushes. According to one resource, the average human throws away over 300 pounds of plastic a year, so at this extreme usage level this would account for 0.08% of their yearly plastic waste, or, in other words, every day they throw away four times the amount of other plastic that they throw away in toothbrushes per year.
In other words, even if you designed an everlasting brush and make it so affordable and effective that adoption goes up past 90%, you've only reduced total plastic waste by 0.08%.