Currently the seed are automaticaly generated in the warehouse with the name dbt_{schema_name}_seed_data, with {schema_name} being the schema name specified in the profiles.yml.
I want to specify a different name, e.g. dbt_processing_seed_data, without changing the schema name in profile.yml to 'processing'.
Reason behind all this, different devs want to have their own schema so they don't interfere with each other. But it is unnecessary that the (same) seed data is stored multiple times in the warehouse.
You can set the schema for a seed in your dbt_project.yml file. See the docs.
To get the behavior you describe, where the target name is not prepended to the schema, you need to override the generate_schema_name macro by creating a new macro with that name in your project. Docs on that are here. You can use the node's resource type so that this behavior is only applied to seeds.
{% macro generate_schema_name(custom_schema_name, node) -%}
{%- set default_schema = target.schema -%}
{%- if custom_schema_name is none -%}{{ default_schema }}
{%- elif node.resource_type == "seed" -%}{{ custom_schema_name | trim }}
{%- else -%}{{ default_schema }}_{{ custom_schema_name | trim }}
{%- endif -%}
{%- endmacro %}
I'd caution against this, though. Seeds are version-controlled, and really aren't intended to be used for large raw datasets (see the docs again). Since they get checked in alongside code, they should really share the same separation of environments that the code has.