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I am using Active Product Filters native Woo Widget which displays all active filters applied to in product loop. Now, I have added filtering by tags and problem appeared.

Filtering by tags is working okay in view of displaying products with selected tag. To filter by tags I am using GET request with product_tag parameter.

The problem is next. Active Filters Widget doesn't display Tag as active filter. For example, when I select Product Size or Price as Filter, this is displayed at Front End as active filter and there is an option to remove that filter, but when Tag is selected Widget doesn't add that Tag as active filter.

My question is how I can edit active filters array in PHP to add filters from product_tag parameter?

The "Active Product Filters" widget takes data from "WC_Query::get_layered_nav_chosen_attributes".

But I did not find a hook there that could be used.

As an variant, create custom widget and extends for "WC_Widget_Layered_Nav_Filters" class and override the "widget" method.

Or try the solution.
For example, there is a plugin https://woocommerce.com/products/product-filters.
Product filters with ajax.
There is a 'Notes for Product Filters' widget that displays active filters and tags (which is what you need).