(Amended per comments)
AFAICT, in Czech, the animate/inanimate distinction is considered as a kind of “sub-gender” specification for the masculine gender. Usually you need to distinguish only m/f/n, but in a few cases, the masculine gender needs to be differentiated to animate/inanimate.
Masculine animate nouns have the singular accusative different from the singular nominative (e.g. man: nom muž, acc muže) while inanimate nouns have accusative identical to nominative (e.g. castle: nom hrad, acc hrad). This is the difference native speakers use to recognize noun animacy (and choose the correct declension paradigm). Furthermore, there are differences in singular dative and locative (e.g. mužovi vs. hradu), and plural nominative (e.g. muži vs. hrady).
The most important aspect affected by animacy is the third-person plural past verb form: the inanimate nouns are treated almost like feminines, e.g. muži byli (“men were”), while hrady byly (similarily to ženy byly = “women were”).
We have four declension paradigms for animate masculine nouns and two for inanimate masculine nouns, however, they are usually listed together and mixed (“pán, hrad, muž, stroj, předseda, soudce”, where hrad and stroj are those inanimate). So, in general usage, the animacy is definitely not on the same level of importance as gender.
As for the “assignment” of animacy to nouns, it is generally true that animate nouns correspond to live beings, while abstract concepts and non-living entities correspond to inanimate nouns. But there are exceptions to this rule. Plants (e.g. strom = tree) and collective nouns (hmyz = insect) are inanimate, and a non-living, yet human-like snowman (sněhulák), or a by-definition-deceased nebožtík are animate.
Then we have several classes of nouns which may be used either as animate or inanimate. Some of them according to the speaker’s style choice: Nouns ending with -tel or -ec (e.g. ledoborec = icebreaker) can use ledoborci vypluli, or ledoborce vypluly (choosing the noun variant and then the corresponding verb inflection), similarly with a few other small sets of nouns.
Then there are nouns which are animate or inanimate according to the specific real-world meaning, e.g. sběrač, which might mean a gatherer (a person gathering something), or a pantograph (a device collecting something), and according to the meaning, it is animate (“sběrači měli plné košíky” – “gatherers had their baskets full”), or inanimate (“sběrače měly poruchu” = “collectors were broken”). An interesting word in this regard is robot (you might be aware that it is originally Czech, invented by Josef Čapek for his brother’s theatrical play R.U.R.): it might be either animate (humanoid, intelligent robots), or inanimate (dumb machines, e.g. a kitchen robot).
For some nouns, you can also use animate declension expressively, e.g. (“dám si ruma” = “I’ll have a rum”, even though rum is normally inanimate, so “dám si rum” is the basic usage).