The concept of "monster Level" is introduced in the Dungeon Master's Guide, Appendix C, page 174. The levels for the creatures in the Monster Manual are in the tables on the following pages of that appendix. The listings in the Monster Manual II include it right in the monster listing for convenience, because the book wasn't published when the Appendix C tables were compiled. As the Monster Manual II says in How To Use This Book (p. 6; emphasis mine):

LEVEL and EXPERIENCE POINT VALUE are determined by the method indicated in THE DUNGEON MASTER'S GUIDE. If a type of monster has varying hit dice and/or experience levels, values for each are shown.

Monster Level is a rough indicator of relative challenge, but is far from an exact science. Monster level is a direct function of the XP value of the creature, as calculated from the table on page 174 of the DMG. Monster level is roughly equivalent to the level of the dungeon on which they will be commonly found, though they can be found lower or higher, in greater or lesser numbers encountered.

Created as a rough guide for random encounters, it should be taken only as a suggestion, and as a means of sorting creatures into any random encounter tables you might create yourself. It was never intended to be used as a fixed measure of difficulty, and if you try to use it that way (like CR in WotC editions of D&D) you'll find your games become more unpredictable instead of more predictable.

As was mentioned in this related question....game designers aren't generally very good at biology. As you can see in my answer to this question, they slapped a mass on a gelatinous cube, and ended up with a creature less dense than cork.

Now, that said, while there are no rules for this laid out in 5E, there are other resources that may be useful to you.

Released as part of the D&D 3.5 SRD, we have a table of creature size and scale that includes weight ranges for creatures of a given Size. This table can also be found at the bottom of this page in the Hypertext SRD, but the first link is a little easier to access.

If you want more specific information on a given creature, many entries in the 3.5E Monster Manuals included a section on typical height and weight. For example....

An Astral Deva is about 7-1/2 feet tall and weighs about 250 lbs.

3.5E MM, page 11.

A Displacer Beast is the size of a Bengal Tiger, about 9 feet long and weighs about 500 lbs

page 67

However, not every entry has this information. For example, it is lacking in the Dragons section.

As a final option, you can do what the first link in this answer recommends, and make use of the Square Cube law to ballpark a weight.

To use this...take a creature that has a similar body plan to your other creature and look at how different they are in height (or length).

$$ Multiplier = NewHeight/OldHeight $$

Now, take the mass of the original creature and...

$$ NewWeight = OldWeight*(Multiplier)^3 $$

Using this, you can put a ballpark on the weight of a creature based on the weight of an RL creature that has a similar build.

Between these three resources, the table, the specific per-creature entries from 3.5E Monster Manuals, and the square-cube law, that should be enough to let you at least ballpark the weight of any creature.