Why is the foreign key part of the primary key in an identifying relationship?

Logically, this would be an identifying relationship, since a form field cannot exist without a form.

No, identifying relationship is about identification, not existence.

Any X:Y relationship where X >= 1 guarantees existence of the left side, whether identifying or not. In your case, a 1:N relationship guarantees existence of form for any given form_field. You could make it identifying or non-identifying and it would still guarantee the same.

Remarks:

• You would model an identifying relationship by making form_field.form_id part of a key. For example form_field PK could look like: {form_id, label}, which BTW would be quite beneficial for proper clustering of your data (InnoDB tables are always clustered).
• Just making a PK: {id, form_id} would be incorrect, since this superkey is not a candidate key (i.e. it is not minimal - we could remove form_id from it and still retain the uniqueness).
• You would model a 0..1:N relationship by making the form_field.form_id NULL-able (but then you wouldn't be able to make it identifying as well - see below).

There are two definitions of the "identifying relationship":

• Strict definition: A relationship that migrates parent key into child primary key¹.
• Loose definition: A relationship that migrates parent key into child key.

In other words, the loose definition allows migration into alternate key as well (and not just primary).

Most tools² seem to use the strict definition though, so if you mark the relationship as identifying, that will automatically make the migrated attributes part of the child PK, and none of the PK attributes can be NULL.

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¹ Which is then either completely comprised from migrated attributes, or is a combination of migrated attributes and some additional attributes.

² ERwin and Visio do. I haven't used MySQL Workbench for modeling yet, but your description seems to suggest it behaves the same.

An identifying relationship is supposed to be one where the primary key includes foreign key attributes. That's why when you designate a relationship as identifying the posted foreign key is deemed to be part of the primary key.

The difference between an "identifying" relationship and a non-identifying one is purely informational or diagrammatic if the same key constraints and nullability constraints apply in each case. The concept is analogous to and a consequence of designating a "primary" key. If a table has more than one candidate key then all other things being equal it doesn't matter from a logical perspective which key is designated the primary one - the form, function and (presumably) the business meaning of the table is the same.

In your example however, the keys in the two tables are NOT the same. In the first case ID is unique in the form_field table while in the second case it apparently isn't. I expect that's not what you intended.