designing database to hold different metadata information

This is called the Observation Pattern.

Three objects, for the example

Book
Title="Gone with the Wind" 
Author="Margaret Mitchell"
ISBN   = '978-1416548898'

Cat
Name="Phoebe"
Color="Gray"
TailLength = 9 'inch'

Beer Bottle
Volume = 500 'ml'
Color="Green"

This is how tables may look like:

Entity
EntityID    Name            Description
   1        'Book'            'To read'
   2        'Cat'             'Fury cat' 
   3        'Beer Bottle'     'To ship beer in'

.

PropertyType
PropertyTypeID   Name        IsTrait         Description
   1            'Height'     'NO'       'For anything that has height' 
   2            'Width'      'NO'       'For anything that has width' 
   3            'Volume'     'NO'       'For things that can have volume'
   4            'Title'      'YES'      'Some stuff has title' 
   5            'Author'     'YES'      'Things can be authored' 
   6            'Color'      'YES'      'Color of things' 
   7            'ISBN'       'YES'      'Books would need this'
   8            'TailLength' 'NO'       'For stuff that has long tails'
   9            'Name'       'YES'      'Name of things'

.

Property
PropertyID   EntityID  PropertyTypeID      
    1           1              4     -- book, title
    2           1              5     -- book, author
    3           1              7     -- book, isbn
    4           2              9     -- cat, name
    5           2              6     -- cat, color
    6           2              8     -- cat, tail length
    7           3              3     -- beer bottle, volume
    8           3              6     -- beer bottle, color

.

Measurement
PropertyID     Unit       Value 
    6          'inch'       9          -- cat, tail length
    7          'ml'        500         -- beer bottle, volume

.

Trait
PropertyID         Value 
    1         'Gone with the Wind'     -- book, title
    2         'Margaret Mitchell'      -- book, author
    3         '978-1416548898'         -- book, isbn
    4         'Phoebe'                 -- cat, name
    5         'Gray'                   -- cat, color
    8         'Green'                  -- beer bottle, color

EDIT:

Jefferey raised a valid point (see comment), so I’ll expand the answer.

The model allows for dynamic (on-fly) creation of any number of entites
with any type of properties without schema changes. Hovewer, this flexibility has a price — storing and searching is slower and more complex than in a usual table design.

Time for an example, but first, to make things easier, I’ll flatten the model into a view.

create view vModel as 
select 
      e.EntityId
    , x.Name  as PropertyName
    , m.Value as MeasurementValue
    , m.Unit
    , t.Value as TraitValue
from Entity           as e
join Property         as p on p.EntityID       = p.EntityID
join PropertyType     as x on x.PropertyTypeId = p.PropertyTypeId
left join Measurement as m on m.PropertyId     = p.PropertyId
left join Trait       as t on t.PropertyId     = p.PropertyId
;

To use Jefferey’s example from the comment

with 
q_00 as ( -- all books
    select EntityID
    from vModel
    where PropertyName="object type"
      and TraitValue="book" 
),
q_01 as ( -- all US books
    select EntityID
    from vModel as a
    join q_00   as b on b.EntityID = a.EntityID
    where PropertyName="publisher country"
      and TraitValue="US" 
),
q_02 as ( -- all US books published in 2008
    select EntityID
    from vModel as a
    join q_01   as b on b.EntityID = a.EntityID
    where PropertyName="year published"
      and MeasurementValue = 2008 
),
q_03 as ( -- all US books published in 2008 not discontinued
    select EntityID
    from vModel as a
    join q_02   as b on b.EntityID = a.EntityID
    where PropertyName="is discontinued"
      and TraitValue="no" 
),
q_04 as ( -- all US books published in 2008 not discontinued that cost less than $50
    select EntityID
    from vModel as a
    join q_03   as b on b.EntityID = a.EntityID
    where PropertyName="price"
      and MeasurementValue < 50 
      and MeasurementUnit="USD"
)
select
      EntityID
    , max(case PropertyName when 'title' than TraitValue else null end) as Title
    , max(case PropertyName when 'ISBN'  than TraitValue else null end) as ISBN
from vModel as a
join q_04   as b on b.EntityID = a.EntityID
group by EntityID ;

This looks complicated to write, but on a closer inspection you may notice a pattern in CTEs.

Now suppose we have a standard fixed schema design where each object property has its own column.
The query would look something like:

select EntityID, Title, ISBN
from vModel
WHERE ObjectType="book"
  and PublisherCountry = 'US'
  and YearPublished    = 2008
  and IsDiscontinued   = 'no'
  and Price            < 50
  and Currency         = 'USD'
;