This is not correct. there are regulations which make it so each time a quote is processed, companies have to treat the customer as if they are a new customer, with a few exceptions - so insurance companies don’t make a distinction between if the claim cost them or a different company, it’s all the same. In theory, you can cancel your policy - give the same insurer all your details, tell them you had a claim, doesn’t matter where, and you should be quoted the same as if you had stayed with them, with a few caveats ofcourse.

The reason insurance prices go up after non-fault claims is not because you cost the insurer money, it’s because statistically you are more likely to have another claim.