Is there a name for this database schema of key values?

Question

We process a routine data feed from a client who just refactored their database from a form that seems familiar (one row per entity, one column per attribute) to one that seems unfamiliar to me (one row per entity per attribute):

Before: one column per attribute

ID   Ht_cm   wt_kg   Age_yr  ... 
1      190      82     43    ...
2      170      60     22    ...
3      205      90     51    ...

After: one column for all attributes

ID    Metric   Value
 1     Ht_cm     190
 1     Wt_kg     82
 1     Age_yr    43
 1      ...
 2     Ht_cm     170
 2     Wt_kg     60
 2     Age_yr    22
 2     ...
 3     Ht_cm     205
 3     Wt_kg     90
 3     Age_yr    51
 3     ...

Is there a name for this database structure? What are the relative advantages? The old way seems easier to place validity constraints on specific attributes (non-null, non-negative, etc.) and easier to calculate averages. But I can see how it might be easier to add new attributes without refactoring the database. Is this a standard/preferred way of structuring data?