With some serious Googling, and with the help of the Community and this gist I was able to successfully get a token from Sitecore 9.1's Identity Server.
First, you need to add a new Client to the Sitecore.IdentityServer.Host.xml file (\Config\production\Sitecore.IdentityServer.Host.xml).
I dug up more details on the PasswordClient and created my own version by copying PasswordClient as a template and called it PostmanClient giving it a client id of "postman-api".
<Clients>
<DefaultClient>
<AllowedCorsOrigins>
<AllowedCorsOriginsGroup1>https://habitathome.dev.local|https://habitathomebasic.dev.local</AllowedCorsOriginsGroup1>
</AllowedCorsOrigins>
</DefaultClient>
<PasswordClient>
<ClientSecrets>
<ClientSecret1>ClientSecret</ClientSecret1>
</ClientSecrets>
</PasswordClient>
<PostmanClient>
<ClientId>postman-api</ClientId>
<ClientName>postman-api</ClientName>
<AccessTokenType>0</AccessTokenType>
<AllowOfflineAccess>true</AllowOfflineAccess>
<AlwaysIncludeUserClaimsInIdToken>false</AlwaysIncludeUserClaimsInIdToken>
<AccessTokenLifetimeInSeconds>3600</AccessTokenLifetimeInSeconds>
<IdentityTokenLifetimeInSeconds>3600</IdentityTokenLifetimeInSeconds>
<AllowAccessTokensViaBrowser>true</AllowAccessTokensViaBrowser>
<RequireConsent>false</RequireConsent>
<RequireClientSecret>true</RequireClientSecret>
<AllowedGrantTypes>
<AllowedGrantType1>password</AllowedGrantType1>
</AllowedGrantTypes>
<AllowedCorsOrigins>
</AllowedCorsOrigins>
<AllowedScopes>
<AllowedScope1>openid</AllowedScope1>
<AllowedScope2>sitecore.profile</AllowedScope2>
<AllowedScope3>sitecore.profile.api</AllowedScope3>
</AllowedScopes>
<ClientSecrets>
<ClientSecret1>ClientSecret</ClientSecret1>
</ClientSecrets>
<UpdateAccessTokenClaimsOnRefresh>true</UpdateAccessTokenClaimsOnRefresh>
</PostmanClient>
</Clients>
Once that was done (and I restarted identity server) I was able to use the gist sample and provide values relevant to my newly added client:
$identityserverUrl = "https://<url-to-your-identityserver>"
$tokenendpointurl = $identityserverUrl + "/connect/token"
$granttype = "password" # client_credentials / password
$client_id = "postman-api"
$client_secret = "ClientSecret"
$username = "sitecore\admin"
$password = "superStrongPassword"
$scope = "openid"
We are then left with an access_token which can be used to connect to Sitecore.